Most recent edit on 2009-04-10 02:30:23 by CharlesFrancis
Additions:
← Foundations of Quantum Theory ↑ →
Quantum theory is often thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Quantum Logic
In mathematics, a logic» is a formal language (from logos, meaning "word" in greek). In a logic, statements, or propositions have a particular form and are given truth values. In classical, Boolean», or two-valued logic, truth values are either 0 or 1, for statements which are FALSE or TRUE respectively. Quantum logic» is a many valued logic». In many valued logics, truth values between 0 and 1 are allowed for statements which are neither true nor false, but have some kind of intuitive level of truth somewhere in between. Fuzzy logic» is an example of a many valued logic, much used in control engineering. Probability theory» is also a many valued logic, described as The Logic of Science», in the classic book» by E. T. Jaynes», and also called Bayesian reasoning».
Quantum logic (due to Birkhoff» and von Neumann») is a language which tells us about possible results of measurements. It does not tell us, in any direct way, what happens between measurements (later, when we are fluent in the language, we can draw some conclusions). Quantum logic is usually treated in a very abstract, and somewhat obscure, way. This treatment aims keep it as concrete as possible and to show how to translate the mathematical symbols of standard quantum theory into statements in the English language.
Feynman» is famously quoted as having said “I think it is safe to say that no one understands quantum mechanics”. The plethora of “interpretations” seems to bear this out. Yet I claim quantum mechanics can be understood. In essence, what I will describe is the “orthodox”, or Dirac-Von Neumann interpretation. If I have contributed anything, it is to make the interpretation easier to understand, not to alter it in a fundamental way. When I read von Neumann on interpretation, I do not think that what he says is really different from what I say. However, von Neumann was one of the most brilliant mathematicians in history. He had a way of talking over one’s head, which is difficult to follow. Mathematics is a language, but the natural ability to treat it like an ordinary language is not really human. This goes beyond the more common ability to manipulate equations according to rules, which does not require understanding. One way to learn language is to start by translating simple phrases and sentences into English from a primer. This section is a primer in the language of quantum logic, as well as an introduction to the mathematical structure of quantum theory.
Quantum logic is often rejected as an interpretation of quantum mechanics on the ground that it consists of obscure truth values for simple propositions. In my view this is wrong. It is better described as intuitive truth values for sophisticated propositions. In order to understand it we need to understand what the propositions of quantum mechanics are, and how to translate them into ordinary language. We will understand that these propositions refer to hypothetical measurements, and are not, in general, strictly true or false, but have levels of truth somewhere in between. In this section I describe the language for measurements of position of a single interacting particle. Later sections will treat states of more than one particle, interactions between particles, and introduce the mysterious property of spin».
Truth Values
Classical logic applies to sets of statements about the real world which are definitely true or definitely false. For example, when we make a statement,
P(x) = “The position of a particle is x”,
we tend to assume that it is definitely true or definitely false. Such statements are said to be sharp, meaning that they have truth values from the set {0,1}. When it is the case that P(x) is definitely either true or false then classical logic and classical mechanics apply.
We cannot say that a statement about the future is strictly true or false.
Q(x) = “When a measurement of position is done, the result will be x”.
We may, however, assign a probability to such a statement. If we consider probabilities as truth values, then probability theory is a many valued logic, applying to sentences in the future tense.
In quantum mechanics we also talk about situations in which there is not going to be a measurement. Hypothetical measurement results can be described using statements in the subjunctive mood»:
R(x) = “If a measurement of position were done, then the result would be x”.
R(x) is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like R(x), in the subjunctive mood.
Sentences Describing Hypothetical Measurement Results
Statements in the subjunctive consist of two clauses, the conditional clause “If a measurement of position were done, …”, and the consequent clause “…, then the result would be x”. Quantum mechanics is based on statements composed of a conditional clause and a consequent clause. To formally describe physics using mathematics, we need to be more precise. The conditional clause must also contain whatever information is known before measurement. This information comes from a prior measurement. We therefore discuss two measurements, the first to determine the condition and the second to determine the outcome, or consequence. We represent the results of these measurements symbolically.
The conditional clause, referring to the first measurement, is represented by a ket. It is described as a formal conditional clause to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The basic conditional clauses, on which the language is built, refer to measurements of position:
RULE I. 
is the formal conditional clause “If measured position were x, …”.
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of
x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing
N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where
N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of
covariance in the quantum domain. This will be considered in
Quantum Covariance. For the immediate development, it will be assumed that
Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to
OR. In grammar
OR is a
coordinating conjunction». In logic and mathematics it is
logical disjunction».
OR will be represented by the symbol
+. A simple logical disjunction does not distinguish the likelihood of the two possibilities;
“If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of
a is greater than that of
b, then
means
“if measured position were either x or y, but more likely x, …”. Precise values for
a and
b in a given situation will be determined as the language develops.
We want to be able to express many possibilities,
“if the particle were found at x or y or z or …”, and we want weighting between all options. This is done recursively, by starting from the basic conditional clauses and repeating the following rule as many times as we require:
RULE II. If 
and 
are formal conditional clauses, and a and b are complex numbers, then 
is a formal conditional clause.
The set of formal conditional clauses, or kets, now has the mathematical structure of an N-dimensional vector space. The symbol H1(t) is used to denote this vector space, and consists of a family of statements we can make about the measurement of position of a single particle at time t. The basic conditional clauses, 
, described in rule I are a basis for H1(t).
Definition: Kets are loosely refered to as states.
Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use “state”, in keeping with common practice when no confusion arises.
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another. Before we can study the evolution of states in time we need to complete the language for talking about measurements of position at a particular time.
A conditional clause on its own has little meaning. To complete a formal sentence in quantum theory we need to put it together with a consequent clause. Consequent causes refer to a second measurement, at the same time as the first measurement. For hypothetical measurements, there is no problem with the idea that they both take place at the same time. To make statements about real measurement results we will also need to know how kets change in time.
There is no fundamental difference between one measurement and another, so the grammatical structure, weighted disjunction described in rule II, applies equally well to consequent clauses. These also form an N-dimensional vector space defined from a basis of consequent clauses in one-one correspondence with the basic conditional clauses, or kets, described by rule I. Consequent clauses are represented symbolically by bras, according to rule III:
RULE III. 
is the formal consequent clause “…, then, in a second measurement at the same time, measured position would be x”.
We put the two clauses together, to make a braket, representing a statement about measurement at a given time:
RULE IV. 
is the statement “If measured position were y, then, in a second measurement at the same time, measured position would be x”.
From observation we know that, if, at some particular time, a particle is measured at position x, then its position is definitely x and it cannot be measured separately at some other position y at the same time. The statement 
is strictly true or false, depending on whether or not x = y. Its truth value is given by a Kronecker delta, and we write:
.
between any two kets, 
and 
in H1(t). Thus, H1(t) is a Hilbert space and the basic states of rule I are an orthonormal basis.
Integral Representation
In the language of quantum theory, absolute magnitude has no meaning; only relative magnitudes are important in weighted logical OR. For any complex number a, the clause 
means exactly the same thing as 
. In other words, normalisation is irrelevant. For large values of N, it is often convenient to normalise basis states so that the inner product 
becomes a Dirac delta function instead of a Kronecker delta,
This normalisation is natural when using the language of integrals rather than the language of sums. If we bear in mind that strictly the integrals stand for finite sums with N terms, everything remains well defined, and we be able to control the divergence problems of quantum electrodynamics». With this normalisation, the resolution of unity,
is replaced by,
The integral is just what was defined previously, but uses three dimensions instead of one. It is regarded strictly as a sum of a N terms where N is a finite number of possible positions, rather than an infinite sum.
Plane Wave States
For a 3-vector, p, at the origin, define a particular ket, 
, as a sum of position states, 
:
Taking the inner product with 
defines a sinusoidal wave,
This function can be seen to have planar wave crests, perpendicular to vector p and at equally spaced intervals 2π/|p|. It is a plane wave» at constant time.
Definition: 
is a plane wave state with momentum p.
Definition: The space of momenta, p, is momentum space.
Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that
p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using
natural units», in which the
Dirac constant»,
. To convert to conventional units, substitute
Definition: For each ket, 
, define the momentum space wave function, f (p), such that
Using the dot product with the laws of indices, the delta function in 3 dimensions is the product of three 1 dimensional delta functions. So,
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the Fourier inversion theorem». In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over p and over x (or y) really stand for finite sums with N terms, where N is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which N goes to infinity, provided that N is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in Quantum Covariance and in Discrete Quantum Electrodynamics.
The Resolution of Unity in Momentum Space
With a little rearrangement,
Using the integral representation of the resolution of unity, substituting for 
, and using the resolution of unity again, for any kets 
and 
,
So, we have another resolution of unity, this time in terms of momentum states:
So, any ket can be written as a sum of plane wave states,
This shows that plane wave states are a basis for Hilbert space. Mathematically, switching between representations in terms of momentum and position is simply a change of basis, analogous to a change of coordinate axes for 3-vectors.
Foundations of Quantum Theory ↑ Observable Quantities →
Deletions:
← Quantum Weirdness ↑ →
Quantum theory is famous for the paradoxical way in which it describes the universe. More strictly the paradoxes lie in the “standard”
Copenhagen interpretaion» of quantum mechanics, not in its mathematical structure. Some physicists adopt
other interpretations», but these appear inadequate to properly describe properties of matter in a complete, coherent, and consistent manner. Many declare that interpretation is not the business of physics. This section contains a general description of quantum theory and of paradox in the Copenhagen interpretation, and describes how the paradoxes are addressed in relational quantum gravity.
Wave Particle Duality
The Copenhagen interpretation» of quantum mechanics invokes the notion of complementarity», or wave-particle duality, according to which the fundamental building blocks of matter are neither wave nor particle, but exhibit the properties of both. When we measure the position of a particle it appears as a point-like object with a precise position in space, but when left to its own devices, it evolves according to the laws of wave mechanics». This raises serious conceptual problems. We understand what a wave is, and we understand what a particle is, but it is difficult to conceive that something can be both wave and particle depending on what an experimental physicist does with it.
Young’s Slits
 The laws of quantum mechanics and of the issues contained in wave particle duality are best illustrated by the Young's slits experiment», originally performed by Thomas Young» in 1801 to resolve the long standing question of whether light was composed of particles, as Newton» had believed, or whether it was a wave, as proposed by Huygens».Young made a point light source by shining sunlight through a single slit. This produces coherent light which passes through a pair of slits and illuminates a screen (in modern versions of the experiment, it is usual to use a laser light source, in which case the single slit is not required). Since the wave theory and the particle theory predict different patterns on the screen, the experiment should determine which is correct. If light consists of particles there should be a bright area in the centre of the screen, getting darker to either side. But if light is a wave then a circular wave pattern emanates from each slit. Where wave crests (red) meet crests and troughs (cyan) meet troughs, the amplitude of the wave increases creating a bright area on the screen. Where crests meet troughs, the waves cancel out (grey), leading to a dark region. These bright and dark bands are called interference fringes. Interference fringes are found, so Young concluded that light is a wave.
Now imagine a reduction in the intensity of the light emitted from the source. At first the interference fringes merely grow dim, but if the intensity is reduced enough, light starts to behave like particles (photons»). For a sufficiently dim light source, only one photon at a time passes through the slits. One photon does not produce interference fringes on screen. Instead it arrives at a precise point on the screen, behaving like a particle. A second photon arrives at a second point. If, over a period of time, a large number of photons pass through the slits, one at a time, the cumulative effect is the interference pattern. There is a large chance that each photon lands on the screen in one of the bright areas, and a very slim chance that it lands in a dark region. This is a most peculiar result, because the photon, which behaves like a particle when it leaves the source and when it arrives at the screen, seems to have the behaviour of a wave when travelling between the two. |
Electrons as Waves
The Young’s slits experiment with single photons was not carried out until modern times, but in 1905 Einstein had shown that the photoelectric effect» could be explained by hypothesising that light comes in quanta, now known as photons. In 1923 de Broglie» reasoned that if a wave (light) turns out to be composed of particles, then particles might also behave like waves. This turned out to be correct. If the Young's slit experiment is performed for electrons instead of light, exactly the same result is obtained – electrons arrive at the screen as particles, each one at a point, but more of them arrive in specific areas. Again, the cumulative effect is an interference pattern.
This raises the questions “Which slit did the particle come through?” and “If it came through a particular slit, how can there be interference in a wave pattern from both slits?”. It is possible to test which slit the electron comes through, for example by shining a light behind the two slits, so that it scatters off the electron just after it passes through them. The electron now behaves like a particle as it comes through the slits. It is always seen to come through one or other slit. But the interference fringes disappear. The measurement to decide which slit the electron came through has changed the overall result. It is easy to see why. Light bounces off each electron as it comes through the slit, effectively kicking it off course and changing the pattern on the screen.
To minimise the effect of shining a light to decide which slit the electron came through, a very dim light might be tried. Now, sure enough, interference fringes start to reappear. But light is composed of photons. If an electron is hit by a photon as it passes through the slit, we can tell which slit the electron came through. The cumulative effect of electrons seen coming through the slits is fringes. But some of the electrons are not hit by a photon, and are not seen at all. The cumulative effect of unseen electrons is the interference pattern.
The Collapse of the Wave Function
Just before the electron hits the screen it apparently takes the form of a wave spread over the region in front of the screen. On the instant it hits the screen, the wave ceases to exist, and the electron is detected at a single point, as though it is a particle. This process is known as the collapse of the wave function», or the reduction of the wave packet.
The two parts of quantum motion:
1. Determinist wave evolution between measurements.
2. Probabilistic collapse in measurement.
There are two parts to motion as described in quantum theory. The first is a determinist, continuous evolution following laws of wave mechanics. The second is a discontinuous, probabilistic change brought about by observation and measurement. This description summarises the whole of quantum theory, so that we may understand the principles of quantum mechanics by understanding the Young’s slits experiment.
The collapse of the wave function takes place instantaneously. As a consequence, special relativity implies that, if the wave is physically real, then according to a moving scientist, part of the wave collapses before the electron hits the screen, and, likewise, the electron becomes a particle before another part of the wave has started to collapse.
The Measurement Problem
The inherent conflict between determinist wave motion and probabilistic collapse has come to be known as the measurement problem». It has been suggested that probabilities in the results of measurement might arise from unknowns within the microscopic structure of a measurement apparatus and that the underlying physical processes in measurement are also governed by a wave equation, but there is no way to reconcile the linear process of wave evolution with the non-linear outcome of measurement. Decoherence» shows how a macroscopic system interacting with a lot of microscopic systems (e.g. the individual light sensitive molecules of a photographic plate) can move from being in a pure quantum state to being in an incoherent mixture of quantum states, but it does not explain how the system comes to being in a particular one of these states.
The Delayed Choice Quantum Eraser
Because light travels faster than electrons, in principle, the electron can pass through the slits before the decision is made, whether to switch on the light or leave it off. If we leave the light off, the electron apparently passes through the slits as a wave. If we switch it on, we find that the electron has passed through one or other slit, as a particle. We can apparently decide what type of behaviour the electron had at the slits after it has passed through them. This simple version of the experiment illustrates the principle, but is not viable in practice. The delayed choice quantum eraser» experiment has been performed, and shows that we actually can determine behaviour at the slits after the particle passes through them.
Schrödinger’s Cat
| Schrödinger» put a cat in a box with a capsule of cyanide, triggered to break with a 50% chance by a quantum mechanical process killing the cat (oh, he didn’t actually do it, but he thought about it). A physicist looking at the box does not know whether the quantum process has broken the capsule or not, so he describes it with a quantum state, that is to say a wave function in which the process has part broken the cyanide capsule, and part not. If the wave function collapses when the observation takes place, then, prior to openning the box, he should describe the cat with a quantum state as well, in which the cat is part alive and desperately trying to get out of the box before the cyanide gets him, and part dead and lying in a heap on the floor. Mouse-over the image to open the box and observe the cat. |
Wigner’s Friend
Wigner» thought about putting his friend in the box, in place of the cat. If the collapse of the wave function does not take place until Wigner opens the box, then what happens to the friend? Is he now a wave function like the cat? If the friend is a physicist, does he not observe the capsule? Why should he not cause the wave function to collapse? What if he is not a physicist? Does anyone have the power to collapse a wave function, even if they don’t know what a wave function is? Can the cat cause the wave function to collapse? Can a beetle in the box collapse the wave function? Or a computer, or a ticker tape timer? Does a physicist outside the door of Wigner’s laboratory describe Wigner as a wave function who has part murdered his friend and part rescued him from the cyanide? Or do we each live in physically different universes, with wave functions collapsing at different times for each of us? These are just some of the issues raised if the wave function is part of the description of the physical properties of matter.
Relational Quantum Gravity
Relational quantum gravity is one of a number of what I call information theoretic interpretations. These interpretations have their roots in the original discussions between Bohr, Heisenberg, and others, which led to the Copenhagen interpretation, but they discard the notion of complementarity. The wave function is not conceived as describing a fundamental property of matter, but rather it describes what we can say about measurement. In these interpretations, the wave function is not real but is simply a way of calculating the probability for the outcome of an experiment. Information theoretic interpretations invert the measurement problem. Collapse is simply the change in a probability once the outcome of a measurement is known, but wave evolution requires explanation.
In my view, the problem with information theoretic interpretations is that they fall short of being complete interpretations of nature. If the wave function describes what we can know about reality, not reality itself, then we are still lacking a description of the underlying physics, and we do not have clear explanations as to why the laws of quantum mechanics yield correct probabilities or why evolution should obey the laws of wave mechanics. These are the issues addressed in relational quantum gravity.
There is no simple explanation of these issues. Rather, we must put together the mathematical treatments of relativity and quantum theory leading to quantum electrodynamics as described in this site. At the root of the explanation is the idea that we should take relativity one step further, into full blooded Cartesian relationism. Not only is motion relative, but position is relative too. Matter is described in terms of fundamental particles. Space and spacetime are not fundamental. They do not apply to the description of an individual particle, but emerge from interactions between many particles. Historically, relationism failed because it could not find a mathematical way in which to describe a universe in which only relative position is possible. Mathematics and physics are now a few generations more advanced. I have sought to show that quantum logic is the natural mathematical language in which we can describe measurement in a relationist universe and that wave evolution is imposed by special relativity.
 If spacetime is an emergent quantity, it can only be used to describe the behaviour of matter when sufficient contact relationships (interactions) exist in the process under study. We can only say which slit a particle comes through if the particle has sufficient contact relationships with other matter to define position with respect to the slits. In special relativity we will define time and space coordinates by the radar method, using two way photon exchange. Later photon exchange will be seen in Feynman diagrams as the process which gives rise to the electromagnetic force and all the structures of matter in our immediate environment. Spacetime structure will be understood in the general case as emergent from the process of photon exchange. Using the radar method, the position coordinate of an event cannot be defined at the time of the event, but only afterwards, when the signal returns. Similarly, the spacial relationship of a particle passing through the slits is not determined at the time at which it passes through the slits, but only later, when the final form of its spacial relationships becomes established in the process of measurement. |
The Einstein Podolsky Rosen Paradox (EPR)
“No reasonable definition of reality can permit this” — Albert Einstein».
“That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to the other, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it” — Isaac Newton».
We observe interference fringes which can be calculated from waves, but there is no direct observation of the wave function or of the infinite speed implicit in collapse. In order to highlight the very deep conflict between relativity and quantum mechanics, Einstein, Rosen and Podolsky imagined that a quantum mechanical process generates two particles flying in opposite directions with equal momenta. The momenta of the particles is not known, so the rules of quantum mechanics dictate that it is governed by a wave function. The two particles become separated. Alice measures the momentum of one particle, and Bob will measure the momentum of the second particle, at a distance remote from Alice. According to conservation of momentum, at the precise time at which Alice does her measurement, the outcome of Bob’s measurement is determined. So, the quantum state of a particle can be instantaneously changed by the remote action of an observer, in apparent conflict with relativity which prevents the instantanteous transmission of an effect.
Einstein believed that some unknown process, such as a hidden variable» (a quantity which can be defined but cannot be known), must dictate the experimental result, but Von Neumann» furnished a proof showing that local hidden variables cannot lead to the result (Bohmian mechanics» seeks to circumvent the problem by postulating non-local hidden variables, but this simply means that instantaneous non-local effects are built into the interpretation from the outset, in defiance of both relativity and common sense).
Bell’s Inequality
John Bell» adapted the EPR experiment to a form which is easier to test experimentally and demonstrated Bell’s inequality», by which the experimental predictions of quantum theory can be tested against theories of local hidden variables. He proposed using a process which emits two particles with equal and opposite spin». Initially spin is not aligned, but it can be measured in any axis. When spin is measured, it aligns with the axis chosen for the measurement. This implies that the other particle must be aligned on the same axis, with opposite spin. It appears that the fact of a spin measurement on one particle not only affects the result of measurement spin of the other, but actually determines the axis on which the spin of the second particle is aligned. Bell’s inequality shows that quantum theory and local hidden variables theories predict different correlations between the results of measurements on different axes, and enables us to experimentally determine which is correct. As stated by Bell, the implication is that, if quantum mechanics is correct, we must sacrifice at least one locality, causality, and realism.
The Aspect Experiment
Bell test experiments» had been carried out a number of times and it had been found that Bell’s inequality is violated; the predictions of quantum mechanics supported. This left the possibility that the wave functions for both particles collapsed at the time of the decision of which axes to use, and that as this took place before the experiment, nothing need to travel faster than the speed of light. Alain Aspect» and his colleagues in Paris set up the experiment in such a way that the decision on which axes were used to measure spin was made by a pseudorandom generator, after the particles were emitted. They still found that Bell’s inequality was violated. Improved versions of the experiment have confirmed the result. The decision on which direction to measure spin of one particle affects the measurement of the other, even though a message from the point of decision to the second particle would have to travel faster than the speed of light.
In spite of this result, no information travels from the results of one measurement to the other. At the time when Bob measures the spin of the second particle, there is no way he can say that his result is affected by Alice’s measurement of the first, because he does not yet know Alice’s result. Alice and Bob’s results must be brought together before a correlation can be found establishing that Bell’s ine quality is violated. The question, however, remains. If nothing travels faster than light, how can the correlation come about?
Locality and Causality
In practice, quantum mechanics is overwhelming supported by experiment. Physics makes no sense if we sacrifice realism. It seems we must have a problem with either locality, causality, or both. In relational quantum gravity we do not have to sacrifice either locality or causality, but we do have to be careful about how we state them. We have to dismiss naive statements based on an assumption of background spacetime. Bell's theorem does not lead us to reject locality or causality, but rather to reject the notion of a fundamental spacetime.
Locality: A particle is in contact with another when it interacts with it. A particle can be considered to be in a neighbourhood of another if, in principle, a photon can be emitted by one particle and absorbed by another, and then a photon emitted by the second and absorbed by the first within a small proper time period.
The reasons for this definition, and the meaning of proper time, will become clear in the treatment of special relativity in the next section. This Cartesian definition reflects the locality condition in qed, and allows that entangled particles in Bell's theorem are separated, in accordance with our intuitive ideas.
Causality: There is a causal relation between two measurements if the outcome of one measurement alters the probability of the outcome the other.
By this definition there is no causal relationship between the measurements of the entangled particles in Bell's theorem. The measurement of one particle does not alter the probabilities for the results of measurement of the other. Only when the two experimenters get together and compare results do they find a correlation. This can only be done at a later time, showing that the correlation is causally related to the measurements, but not that the measurements are causally related to each other.
Emergence of Spacetime
In relational quantum gravity, the resolution of Bell's theorem is not that we must reject realism, locality or causality, but rather we must recognise that spacetime is an emergent concept. It will be found in
quantum electrodynamics that the process of photon exchange is responsible for the electromagnetic force, and all the structures we observe in our immediate environment. In relational quantum gravity we interpret spacetime as emerging as from photon exchange. An electron passing through the slits does not interact with the environment, and does not participate in the structure of space time created by other matter in the environment. It is therefore impossible to say which slit the electron passes through. It remains to understand interference effects which arise from
quantum theory.
The notion of separation can only be said to hold when spacetime exists, that is to say *after* the required physical processes have taken place for the emergence of spacetime. This has not happened at the time of Alice and Bob’s measurements in the
Bell tests, but it has happened when Alice and Bob get together and determine the correlation. There can be no exchange of photons between the immediate environments of Alice and Bob at the time of their measurements, because this would require that photons travel faster than the speed of light. Therefore, while Alice and Bob each observe spacetime structure in their immediate environment, the structure connecting those two regions is not yet complete. At the time when Alice and Bob bring their measurement results together, there will have been many more billions of interactions exchanging photons, and a single spacetime structure containing the regions of spacetime in which Alice and Bob carry out their measurements is then completed.
We cannot say that the outcome of one measurement has effected the other, because this presupposes the existence of spacetime, and puts the logical cart before the horse. We can only discuss the relative orientation of Alice & Bob's measurements in the context of a spacetime structure which contains both. In special relativity the structure of spacetime emerges from two way photon exchange in radar. In relational quantum gravity it emerges from photon exchange in the fundamental structures of matter. We can only talk of spacetime after two way photon exchange has taken place in such a way that spacetime structure has emerged. At the time the measurements take place, spacetime structure is not yet complete.
Quantum Weirdness ↑ Special Relativity →
Edited on 2009-04-03 06:53:38 by CharlesFrancis
Deletions:
Quantum Weirdness ↑ Special Relativity →]] is the natural mathematical language in which we can describe measurement in a relationist universe and that wave evolution is imposed by special relativity.
The notion of separation can only be said to hold when spacetime exists, that is to say *after* the required physical processes have taken place for the emergence of spacetime. This has not happened at the time of Alice and Bob’s measurements in the Bell tests, but it has happened when Alice and Bob get together and determine the correlation. There can be no exchange of photons between the immediate environments of Alice and Bob at the time of their measurements, because this would require that photons travel faster than the speed of light. Therefore, while Alice and Bob each observe spacetime structure in their immediate environment, the structure connecting those two regions is not yet complete. At the time when Alice and Bob bring their measurement results together, there
will have been many more billions of interactions exchanging photons, and a single spacetime structure containing the regions of spacetime in which Alice and Bob carry out their measurements is then completed.
Edited on 2009-04-03 06:52:52 by CharlesFrancis
Additions:
The notion of separation can only be said to hold when spacetime exists, that is to say *after* the required physical processes have taken place for the emergence of spacetime. This has not happened at the time of Alice and Bob’s measurements in the Bell tests, but it has happened when Alice and Bob get together and determine the correlation. There can be no exchange of photons between the immediate environments of Alice and Bob at the time of their measurements, because this would require that photons travel faster than the speed of light. Therefore, while Alice and Bob each observe spacetime structure in their immediate environment, the structure connecting those two regions is not yet complete. At the time when Alice and Bob bring their measurement results together, there will have been many more billions of interactions exchanging photons, and a single spacetime structure containing the regions of spacetime in which Alice and Bob carry out their measurements is then completed.
Quantum Weirdness ↑ Special Relativity →]] is the natural mathematical language in which we can describe measurement in a relationist universe and that wave evolution is imposed by special relativity.
Edited on 2009-04-03 06:51:02 by CharlesFrancis
Additions:
← Quantum Weirdness ↑ →
Quantum theory is famous for the paradoxical way in which it describes the universe. More strictly the paradoxes lie in the “standard”
Copenhagen interpretaion» of quantum mechanics, not in its mathematical structure. Some physicists adopt
other interpretations», but these appear inadequate to properly describe properties of matter in a complete, coherent, and consistent manner. Many declare that interpretation is not the business of physics. This section contains a general description of quantum theory and of paradox in the Copenhagen interpretation, and describes how the paradoxes are addressed in relational quantum gravity.
Wave Particle Duality
The Copenhagen interpretation» of quantum mechanics invokes the notion of complementarity», or wave-particle duality, according to which the fundamental building blocks of matter are neither wave nor particle, but exhibit the properties of both. When we measure the position of a particle it appears as a point-like object with a precise position in space, but when left to its own devices, it evolves according to the laws of wave mechanics». This raises serious conceptual problems. We understand what a wave is, and we understand what a particle is, but it is difficult to conceive that something can be both wave and particle depending on what an experimental physicist does with it.
Young’s Slits
The laws of quantum mechanics and of the issues contained in wave particle duality are best illustrated by the Young's slits experiment», originally performed by Thomas Young» in 1801 to resolve the long standing question of whether light was composed of particles, as Newton» had believed, or whether it was a wave, as proposed by Huygens».Young made a point light source by shining sunlight through a single slit. This produces coherent light which passes through a pair of slits and illuminates a screen (in modern versions of the experiment, it is usual to use a laser light source, in which case the single slit is not required). Since the wave theory and the particle theory predict different patterns on the screen, the experiment should determine which is correct. If light consists of particles there should be a bright area in the centre of the screen, getting darker to either side. But if light is a wave then a circular wave pattern emanates from each slit. Where wave crests (red) meet crests and troughs (cyan) meet troughs, the amplitude of the wave increases creating a bright area on the screen. Where crests meet troughs, the waves cancel out (grey), leading to a dark region. These bright and dark bands are called interference fringes. Interference fringes are found, so Young concluded that light is a wave.
Now imagine a reduction in the intensity of the light emitted from the source. At first the interference fringes merely grow dim, but if the intensity is reduced enough, light starts to behave like particles (photons»). For a sufficiently dim light source, only one photon at a time passes through the slits. One photon does not produce interference fringes on screen. Instead it arrives at a precise point on the screen, behaving like a particle. A second photon arrives at a second point. If, over a period of time, a large number of photons pass through the slits, one at a time, the cumulative effect is the interference pattern. There is a large chance that each photon lands on the screen in one of the bright areas, and a very slim chance that it lands in a dark region. This is a most peculiar result, because the photon, which behaves like a particle when it leaves the source and when it arrives at the screen, seems to have the behaviour of a wave when travelling between the two. |
Electrons as Waves
The Young’s slits experiment with single photons was not carried out until modern times, but in 1905 Einstein had shown that the photoelectric effect» could be explained by hypothesising that light comes in quanta, now known as photons. In 1923 de Broglie» reasoned that if a wave (light) turns out to be composed of particles, then particles might also behave like waves. This turned out to be correct. If the Young's slit experiment is performed for electrons instead of light, exactly the same result is obtained – electrons arrive at the screen as particles, each one at a point, but more of them arrive in specific areas. Again, the cumulative effect is an interference pattern.
This raises the questions “Which slit did the particle come through?” and “If it came through a particular slit, how can there be interference in a wave pattern from both slits?”. It is possible to test which slit the electron comes through, for example by shining a light behind the two slits, so that it scatters off the electron just after it passes through them. The electron now behaves like a particle as it comes through the slits. It is always seen to come through one or other slit. But the interference fringes disappear. The measurement to decide which slit the electron came through has changed the overall result. It is easy to see why. Light bounces off each electron as it comes through the slit, effectively kicking it off course and changing the pattern on the screen.
To minimise the effect of shining a light to decide which slit the electron came through, a very dim light might be tried. Now, sure enough, interference fringes start to reappear. But light is composed of photons. If an electron is hit by a photon as it passes through the slit, we can tell which slit the electron came through. The cumulative effect of electrons seen coming through the slits is fringes. But some of the electrons are not hit by a photon, and are not seen at all. The cumulative effect of unseen electrons is the interference pattern.
The Collapse of the Wave Function
Just before the electron hits the screen it apparently takes the form of a wave spread over the region in front of the screen. On the instant it hits the screen, the wave ceases to exist, and the electron is detected at a single point, as though it is a particle. This process is known as the collapse of the wave function», or the reduction of the wave packet.
The two parts of quantum motion:
1. Determinist wave evolution between measurements.
2. Probabilistic collapse in measurement.
There are two parts to motion as described in quantum theory. The first is a determinist, continuous evolution following laws of wave mechanics. The second is a discontinuous, probabilistic change brought about by observation and measurement. This description summarises the whole of quantum theory, so that we may understand the principles of quantum mechanics by understanding the Young’s slits experiment.
The collapse of the wave function takes place instantaneously. As a consequence, special relativity implies that, if the wave is physically real, then according to a moving scientist, part of the wave collapses before the electron hits the screen, and, likewise, the electron becomes a particle before another part of the wave has started to collapse.
The Measurement Problem
The inherent conflict between determinist wave motion and probabilistic collapse has come to be known as the measurement problem». It has been suggested that probabilities in the results of measurement might arise from unknowns within the microscopic structure of a measurement apparatus and that the underlying physical processes in measurement are also governed by a wave equation, but there is no way to reconcile the linear process of wave evolution with the non-linear outcome of measurement. Decoherence» shows how a macroscopic system interacting with a lot of microscopic systems (e.g. the individual light sensitive molecules of a photographic plate) can move from being in a pure quantum state to being in an incoherent mixture of quantum states, but it does not explain how the system comes to being in a particular one of these states.
The Delayed Choice Quantum Eraser
Because light travels faster than electrons, in principle, the electron can pass through the slits before the decision is made, whether to switch on the light or leave it off. If we leave the light off, the electron apparently passes through the slits as a wave. If we switch it on, we find that the electron has passed through one or other slit, as a particle. We can apparently decide what type of behaviour the electron had at the slits after it has passed through them. This simple version of the experiment illustrates the principle, but is not viable in practice. The delayed choice quantum eraser» experiment has been performed, and shows that we actually can determine behaviour at the slits after the particle passes through them.
Schrödinger’s Cat
| Schrödinger» put a cat in a box with a capsule of cyanide, triggered to break with a 50% chance by a quantum mechanical process killing the cat (oh, he didn’t actually do it, but he thought about it). A physicist looking at the box does not know whether the quantum process has broken the capsule or not, so he describes it with a quantum state, that is to say a wave function in which the process has part broken the cyanide capsule, and part not. If the wave function collapses when the observation takes place, then, prior to openning the box, he should describe the cat with a quantum state as well, in which the cat is part alive and desperately trying to get out of the box before the cyanide gets him, and part dead and lying in a heap on the floor. Mouse-over the image to open the box and observe the cat. |
Wigner’s Friend
Wigner» thought about putting his friend in the box, in place of the cat. If the collapse of the wave function does not take place until Wigner opens the box, then what happens to the friend? Is he now a wave function like the cat? If the friend is a physicist, does he not observe the capsule? Why should he not cause the wave function to collapse? What if he is not a physicist? Does anyone have the power to collapse a wave function, even if they don’t know what a wave function is? Can the cat cause the wave function to collapse? Can a beetle in the box collapse the wave function? Or a computer, or a ticker tape timer? Does a physicist outside the door of Wigner’s laboratory describe Wigner as a wave function who has part murdered his friend and part rescued him from the cyanide? Or do we each live in physically different universes, with wave functions collapsing at different times for each of us? These are just some of the issues raised if the wave function is part of the description of the physical properties of matter.
Relational Quantum Gravity
Relational quantum gravity is one of a number of what I call information theoretic interpretations. These interpretations have their roots in the original discussions between Bohr, Heisenberg, and others, which led to the Copenhagen interpretation, but they discard the notion of complementarity. The wave function is not conceived as describing a fundamental property of matter, but rather it describes what we can say about measurement. In these interpretations, the wave function is not real but is simply a way of calculating the probability for the outcome of an experiment. Information theoretic interpretations invert the measurement problem. Collapse is simply the change in a probability once the outcome of a measurement is known, but wave evolution requires explanation.
In my view, the problem with information theoretic interpretations is that they fall short of being complete interpretations of nature. If the wave function describes what we can know about reality, not reality itself, then we are still lacking a description of the underlying physics, and we do not have clear explanations as to why the laws of quantum mechanics yield correct probabilities or why evolution should obey the laws of wave mechanics. These are the issues addressed in relational quantum gravity.
There is no simple explanation of these issues. Rather, we must put together the mathematical treatments of relativity and quantum theory leading to quantum electrodynamics as described in this site. At the root of the explanation is the idea that we should take relativity one step further, into full blooded Cartesian relationism. Not only is motion relative, but position is relative too. Matter is described in terms of fundamental particles. Space and spacetime are not fundamental. They do not apply to the description of an individual particle, but emerge from interactions between many particles. Historically, relationism failed because it could not find a mathematical way in which to describe a universe in which only relative position is possible. Mathematics and physics are now a few generations more advanced. I have sought to show that quantum logic is the natural mathematical language in which we can describe measurement in a relationist universe and that wave evolution is imposed by special relativity.
| If spacetime is an emergent quantity, it can only be used to describe the behaviour of matter when sufficient contact relationships (interactions) exist in the process under study. We can only say which slit a particle comes through if the particle has sufficient contact relationships with other matter to define position with respect to the slits. In special relativity we will define time and space coordinates by the radar method, using two way photon exchange. Later photon exchange will be seen in Feynman diagrams as the process which gives rise to the electromagnetic force and all the structures of matter in our immediate environment. Spacetime structure will be understood in the general case as emergent from the process of photon exchange. Using the radar method, the position coordinate of an event cannot be defined at the time of the event, but only afterwards, when the signal returns. Similarly, the spacial relationship of a particle passing through the slits is not determined at the time at which it passes through the slits, but only later, when the final form of its spacial relationships becomes established in the process of measurement. |
The Einstein Podolsky Rosen Paradox (EPR)
“No reasonable definition of reality can permit this” — Albert Einstein».
“That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to the other, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it” — Isaac Newton».
We observe interference fringes which can be calculated from waves, but there is no direct observation of the wave function or of the infinite speed implicit in collapse. In order to highlight the very deep conflict between relativity and quantum mechanics, Einstein, Rosen and Podolsky imagined that a quantum mechanical process generates two particles flying in opposite directions with equal momenta. The momenta of the particles is not known, so the rules of quantum mechanics dictate that it is governed by a wave function. The two particles become separated. Alice measures the momentum of one particle, and Bob will measure the momentum of the second particle, at a distance remote from Alice. According to conservation of momentum, at the precise time at which Alice does her measurement, the outcome of Bob’s measurement is determined. So, the quantum state of a particle can be instantaneously changed by the remote action of an observer, in apparent conflict with relativity which prevents the instantanteous transmission of an effect.
Einstein believed that some unknown process, such as a hidden variable» (a quantity which can be defined but cannot be known), must dictate the experimental result, but Von Neumann» furnished a proof showing that local hidden variables cannot lead to the result (Bohmian mechanics» seeks to circumvent the problem by postulating non-local hidden variables, but this simply means that instantaneous non-local effects are built into the interpretation from the outset, in defiance of both relativity and common sense).
Bell’s Inequality
John Bell» adapted the EPR experiment to a form which is easier to test experimentally and demonstrated Bell’s inequality», by which the experimental predictions of quantum theory can be tested against theories of local hidden variables. He proposed using a process which emits two particles with equal and opposite spin». Initially spin is not aligned, but it can be measured in any axis. When spin is measured, it aligns with the axis chosen for the measurement. This implies that the other particle must be aligned on the same axis, with opposite spin. It appears that the fact of a spin measurement on one particle not only affects the result of measurement spin of the other, but actually determines the axis on which the spin of the second particle is aligned. Bell’s inequality shows that quantum theory and local hidden variables theories predict different correlations between the results of measurements on different axes, and enables us to experimentally determine which is correct. As stated by Bell, the implication is that, if quantum mechanics is correct, we must sacrifice at least one locality, causality, and realism.
The Aspect Experiment
Bell test experiments» had been carried out a number of times and it had been found that Bell’s inequality is violated; the predictions of quantum mechanics supported. This left the possibility that the wave functions for both particles collapsed at the time of the decision of which axes to use, and that as this took place before the experiment, nothing need to travel faster than the speed of light. Alain Aspect» and his colleagues in Paris set up the experiment in such a way that the decision on which axes were used to measure spin was made by a pseudorandom generator, after the particles were emitted. They still found that Bell’s inequality was violated. Improved versions of the experiment have confirmed the result. The decision on which direction to measure spin of one particle affects the measurement of the other, even though a message from the point of decision to the second particle would have to travel faster than the speed of light.
In spite of this result, no information travels from the results of one measurement to the other. At the time when Bob measures the spin of the second particle, there is no way he can say that his result is affected by Alice’s measurement of the first, because he does not yet know Alice’s result. Alice and Bob’s results must be brought together before a correlation can be found establishing that Bell’s ine quality is violated. The question, however, remains. If nothing travels faster than light, how can the correlation come about?
Locality and Causality
In practice, quantum mechanics is overwhelming supported by experiment. Physics makes no sense if we sacrifice realism. It seems we must have a problem with either locality, causality, or both. In relational quantum gravity we do not have to sacrifice either locality or causality, but we do have to be careful about how we state them. We have to dismiss naive statements based on an assumption of background spacetime. Bell's theorem does not lead us to reject locality or causality, but rather to reject the notion of a fundamental spacetime.
Locality: A particle is in contact with another when it interacts with it. A particle can be considered to be in a neighbourhood of another if, in principle, a photon can be emitted by one particle and absorbed by another, and then a photon emitted by the second and absorbed by the first within a small proper time period.
The reasons for this definition, and the meaning of proper time, will become clear in the treatment of special relativity in the next section. This Cartesian definition reflects the locality condition in qed, and allows that entangled particles in Bell's theorem are separated, in accordance with our intuitive ideas.
Causality: There is a causal relation between two measurements if the outcome of one measurement alters the probability of the outcome the other.
By this definition there is no causal relationship between the measurements of the entangled particles in Bell's theorem. The measurement of one particle does not alter the probabilities for the results of measurement of the other. Only when the two experimenters get together and compare results do they find a correlation. This can only be done at a later time, showing that the correlation is causally related to the measurements, but not that the measurements are causally related to each other.
Emergence of Spacetime
In relational quantum gravity, the resolution of Bell's theorem is not that we must reject realism, locality or causality, but rather we must recognise that spacetime is an emergent concept. It will be found in
quantum electrodynamics that the process of photon exchange is responsible for the electromagnetic force, and all the structures we observe in our immediate environment. In relational quantum gravity we interpret spacetime as emerging as from photon exchange. An electron passing through the slits does not interact with the environment, and does not participate in the structure of space time created by other matter in the environment. It is therefore impossible to say which slit the electron passes through. It remains to understand interference effects which arise from
quantum theory.
The notion of separation can only be said to hold when spacetime exists, that is to say *after* the required physical processes have taken place for the emergence of spacetime. This has not happened at the time of Alice and Bob’s measurements in the
Bell tests, but it has happened when Alice and Bob get together and determine the correlation. There can be no exchange of photons between the immediate environments of Alice and Bob at the time of their measurements, because this would require that photons travel faster than the speed of light. Therefore, while Alice and Bob each observe spacetime structure in their immediate environment, the structure connecting those two regions is not yet complete. At the time when Alice and Bob bring their measurement results together, there
will have been many more billions of interactions exchanging photons, and a single spacetime structure containing the regions of spacetime in which Alice and Bob carry out their measurements is then completed.
We cannot say that the outcome of one measurement has effected the other, because this presupposes the existence of spacetime, and puts the logical cart before the horse. We can only discuss the relative orientation of Alice & Bob's measurements in the context of a spacetime structure which contains both. In special relativity the structure of spacetime emerges from two way photon exchange in radar. In relational quantum gravity it emerges from photon exchange in the fundamental structures of matter. We can only talk of spacetime after two way photon exchange has taken place in such a way that spacetime structure has emerged. At the time the measurements take place, spacetime structure is not yet complete.
Quantum Weirdness ↑ Special Relativity →
Deletions:
← Foundations of Quantum Theory ↑ →
Quantum theory is often thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Quantum Logic
In mathematics, a logic» is a formal language (from logos, meaning "word" in greek). In a logic, statements, or propositions have a particular form and are given truth values. In classical, Boolean», or two-valued logic, truth values are either 0 or 1, for statements which are FALSE or TRUE respectively. Quantum logic» is a many valued logic». In many valued logics, truth values between 0 and 1 are allowed for statements which are neither true nor false, but have some kind of intuitive level of truth somewhere in between. Fuzzy logic» is an example of a many valued logic, much used in control engineering. Probability theory» is also a many valued logic, described as The Logic of Science», in the classic book» by E. T. Jaynes», and also called Bayesian reasoning».
Quantum logic (due to Birkhoff» and von Neumann») is a language which tells us about possible results of measurements. It does not tell us, in any direct way, what happens between measurements (later, when we are fluent in the language, we can draw some conclusions). Quantum logic is usually treated in a very abstract, and somewhat obscure, way. This treatment aims keep it as concrete as possible and to show how to translate the mathematical symbols of standard quantum theory into statements in the English language.
Feynman» is famously quoted as having said “I think it is safe to say that no one understands quantum mechanics”. The plethora of “interpretations” seems to bear this out. Yet I claim quantum mechanics can be understood. In essence, what I will describe is the “orthodox”, or Dirac-Von Neumann interpretation. If I have contributed anything, it is to make the interpretation easier to understand, not to alter it in a fundamental way. When I read von Neumann on interpretation, I do not think that what he says is really different from what I say. However, von Neumann was one of the most brilliant mathematicians in history. He had a way of talking over one’s head, which is difficult to follow. Mathematics is a language, but the natural ability to treat it like an ordinary language is not really human. This goes beyond the more common ability to manipulate equations according to rules, which does not require understanding. One way to learn language is to start by translating simple phrases and sentences into English from a primer. This section is a primer in the language of quantum logic, as well as an introduction to the mathematical structure of quantum theory.
Quantum logic is often rejected as an interpretation of quantum mechanics on the ground that it consists of obscure truth values for simple propositions. In my view this is wrong. It is better described as intuitive truth values for sophisticated propositions. In order to understand it we need to understand what the propositions of quantum mechanics are, and how to translate them into ordinary language. We will understand that these propositions refer to hypothetical measurements, and are not, in general, strictly true or false, but have levels of truth somewhere in between. In this section I describe the language for measurements of position of a single interacting particle. Later sections will treat states of more than one particle, interactions between particles, and introduce the mysterious property of spin».
Truth Values
Classical logic applies to sets of statements about the real world which are definitely true or definitely false. For example, when we make a statement,
P(x) = “The position of a particle is x”,
we tend to assume that it is definitely true or definitely false. Such statements are said to be sharp, meaning that they have truth values from the set {0,1}. When it is the case that P(x) is definitely either true or false then classical logic and classical mechanics apply.
We cannot say that a statement about the future is strictly true or false.
Q(x) = “When a measurement of position is done, the result will be x”.
We may, however, assign a probability to such a statement. If we consider probabilities as truth values, then probability theory is a many valued logic, applying to sentences in the future tense.
In quantum mechanics we also talk about situations in which there is not going to be a measurement. Hypothetical measurement results can be described using statements in the subjunctive mood»:
R(x) = “If a measurement of position were done, then the result would be x”.
R(x) is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like R(x), in the subjunctive mood.
Sentences Describing Hypothetical Measurement Results
Statements in the subjunctive consist of two clauses, the conditional clause “If a measurement of position were done, …”, and the consequent clause “…, then the result would be x”. Quantum mechanics is based on statements composed of a conditional clause and a consequent clause. To formally describe physics using mathematics, we need to be more precise. The conditional clause must also contain whatever information is known before measurement. This information comes from a prior measurement. We therefore discuss two measurements, the first to determine the condition and the second to determine the outcome, or consequence. We represent the results of these measurements symbolically.
The conditional clause, referring to the first measurement, is represented by a ket. It is described as a formal conditional clause to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The basic conditional clauses, on which the language is built, refer to measurements of position:
RULE I. is the formal conditional clause “If measured position were x, …”.
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of
x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing
N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where
N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of
covariance in the quantum domain. This will be considered in
Quantum Covariance. For the immediate development, it will be assumed that
Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to
OR. In grammar
OR is a
coordinating conjunction». In logic and mathematics it is
logical disjunction».
OR will be represented by the symbol
+. A simple logical disjunction does not distinguish the likelihood of the two possibilities;
“If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of
a is greater than that of
b, then
means
“if measured position were either x or y, but more likely x, …”. Precise values for
a and
b in a given situation will be determined as the language develops.
We want to be able to express many possibilities,
“if the particle were found at x or y or z or …”, and we want weighting between all options. This is done recursively, by starting from the basic conditional clauses and repeating the following rule as many times as we require:
RULE II. If and are formal conditional clauses, and a and b are complex numbers, then is a formal conditional clause.
The set of formal conditional clauses, or kets, now has the mathematical structure of an N-dimensional vector space. The symbol H1(t) is used to denote this vector space, and consists of a family of statements we can make about the measurement of position of a single particle at time t. The basic conditional clauses, , described in rule I are a basis for H1(t).
Definition: Kets are loosely refered to as states.
Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use “state”, in keeping with common practice when no confusion arises.
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another. Before we can study the evolution of states in time we need to complete the language for talking about measurements of position at a particular time.
A conditional clause on its own has little meaning. To complete a formal sentence in quantum theory we need to put it together with a consequent clause. Consequent causes refer to a second measurement, at the same time as the first measurement. For hypothetical measurements, there is no problem with the idea that they both take place at the same time. To make statements about real measurement results we will also need to know how kets change in time.
There is no fundamental difference between one measurement and another, so the grammatical structure, weighted disjunction described in rule II, applies equally well to consequent clauses. These also form an N-dimensional vector space defined from a basis of consequent clauses in one-one correspondence with the basic conditional clauses, or kets, described by rule I. Consequent clauses are represented symbolically by bras, according to rule III:
RULE III. is the formal consequent clause “…, then, in a second measurement at the same time, measured position would be x”.
We put the two clauses together, to make a braket, representing a statement about measurement at a given time:
RULE IV. is the statement “If measured position were y, then, in a second measurement at the same time, measured position would be x”.
From observation we know that, if, at some particular time, a particle is measured at position x, then its position is definitely x and it cannot be measured separately at some other position y at the same time. The statement is strictly true or false, depending on whether or not x = y. Its truth value is given by a Kronecker delta, and we write:
.
Integral Representation
In the language of quantum theory, absolute magnitude has no meaning; only relative magnitudes are important in weighted logical OR. For any complex number a, the clause means exactly the same thing as . In other words, normalisation is irrelevant. For large values of N, it is often convenient to normalise basis states so that the inner product becomes a Dirac delta function instead of a Kronecker delta,
This normalisation is natural when using the language of integrals rather than the language of sums. If we bear in mind that strictly the integrals stand for finite sums with N terms, everything remains well defined, and we be able to control the divergence problems of quantum electrodynamics». With this normalisation, the resolution of unity,
is replaced by,
The integral is just what was defined previously, but uses three dimensions instead of one. It is regarded strictly as a sum of a N terms where N is a finite number of possible positions, rather than an infinite sum.
Plane Wave States
For a 3-vector, p, at the origin, define a particular ket, , as a sum of position states, :
Taking the inner product with defines a sinusoidal wave,
This function can be seen to have planar wave crests, perpendicular to vector p and at equally spaced intervals 2π/|p|. It is a plane wave» at constant time.
Definition: is a plane wave state with momentum p.
Definition: The space of momenta, p, is momentum space.
Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that
p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using
natural units», in which the
Dirac constant»,
. To convert to conventional units, substitute
Definition: For each ket, , define the momentum space wave function, f (p), such that
Using the dot product with the laws of indices, the delta function in 3 dimensions is the product of three 1 dimensional delta functions. So,
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the Fourier inversion theorem». In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over p and over x (or y) really stand for finite sums with N terms, where N is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which N goes to infinity, provided that N is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in Quantum Covariance and in Discrete Quantum Electrodynamics.
The Resolution of Unity in Momentum Space
With a little rearrangement,
Using the integral representation of the resolution of unity, substituting for , and using the resolution of unity again, for any kets and ,
So, we have another resolution of unity, this time in terms of momentum states:
So, any ket can be written as a sum of plane wave states,
This shows that plane wave states are a basis for Hilbert space. Mathematically, switching between representations in terms of momentum and position is simply a change of basis, analogous to a change of coordinate axes for 3-vectors.
Foundations of Quantum Theory ↑ Observable Quantities →
Edited on 2008-09-07 03:02:18 by CharlesFrancis
Additions:
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another. Before we can study the evolution of states in time we need to complete the language for talking about measurements of position at a particular time.
Deletions:
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another, for example, when we want to talk about motion. Before we can study that we need to complete the language for talking about measurements of position at a particular time.
Edited on 2008-08-13 03:33:32 by CharlesFrancis
Additions:
Quantum theory is often thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Deletions:
Quantum theory is thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Edited on 2008-08-13 03:32:27 by CharlesFrancis
Additions:
Quantum theory is thought to be conceptually incomprehensible as physical theory, but it is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not directly to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Deletions:
Quantum theory is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Edited on 2008-08-13 03:30:39 by CharlesFrancis
Additions:
Quantum theory is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not to physical reality. All that is required to understand it is a bit of mathematical trickery applied to a language describing general principles of measurement.
Deletions:
Quantum theory is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not to physical reality. All that is required is a bit of mathematical trickery applied to a language describing general principles of measurement.
Edited on 2008-08-13 03:25:29 by CharlesFrancis
Additions:
← Foundations of Quantum Theory ↑ →
Deletions:
← Foundations of Quantum Theory →
Edited on 2008-08-09 00:20:04 by CharlesFrancis
Additions:
Definition: Kets are loosely refered to as states.
Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use “state”, in keeping with common practice when no confusion arises.
Deletions:
Definition: Kets are loosely refered to as states
Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use state, in keeping with common practice when no confusion arises.
Edited on 2008-08-09 00:18:06 by CharlesFrancis
Additions:
Definition: Kets are loosely refered to as states
Kets are often known as states. They are not strictly states of a particle, but formal conditional clauses describing the likelihood of particular measurement results. That is something of a mouthful. I will use state, in keeping with common practice when no confusion arises.
Deletions:
Definition: Kets are loosely refered to as states
Kets are often known as states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
Edited on 2008-08-09 00:15:33 by CharlesFrancis
Additions:
Definition: Kets are loosely refered to as states
Kets are often known as states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
Deletions:
DefinitionKets are loosely refered to as states
Kets are often known as states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
Edited on 2008-08-09 00:14:20 by CharlesFrancis
Additions:
DefinitionKets are loosely refered to as states
Kets are often known as states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another, for example, when we want to talk about motion. Before we can study that we need to complete the language for talking about measurements of position at a particular time.
Deletions:
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another, for example, when we want to talk about motion. Before we can study that we need to complete the language for talking about measurements of position at a particular time. Kets are often known as states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
Edited on 2008-05-05 01:08:33 by CharlesFrancis
Additions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using natural units», in which the Dirac constant», . To convert to conventional units, substitute
Deletions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using natural units», in which the Dirac constant», . To convert to conventional units, substitute
Edited on 2008-03-13 04:44:29 by CharlesFrancis
Additions:
An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. A simple logical disjunction does not distinguish the likelihood of the two possibilities; “If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of a is greater than that of b, then means “if measured position were either x or y, but more likely x, …”. Precise values for a and b in a given situation will be determined as the language develops.
Deletions:
An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +.
A simple logical disjunction does not distinguish the likelihood of the two possibilities; “If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of a is greater than that of b, then means “if measured position were either x or y, but more likely x, …”. Precise values for a and b in a given situation will be determined as the language develops.
Edited on 2008-03-13 04:43:34 by CharlesFrancis
Additions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +.
Deletions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Edited on 2008-03-13 04:39:37 by CharlesFrancis
Additions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Deletions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Edited on 2008-03-13 04:34:49 by CharlesFrancis
Additions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Deletions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Edited on 2008-03-13 04:33:24 by CharlesFrancis
Additions:
Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Deletions:
<img title="A discrete lattice is chosen from which to define position kets" alt="QuantumCovariance-1" src="images/quantumcovariance/QuantumCovariance-1.gif"> Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to OR. In grammar OR is a coordinating conjunction». In logic and mathematics it is logical disjunction». OR will be represented by the symbol +. The use of a discrete lattice has implications to the mathematical statement of covariance in the quantum domain]]. This will be considered in Quantum Covariance. For the immediate development, it will be assumed that Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
Oldest known version of this page was edited on 2008-03-13 04:29:53 by CharlesFrancis []
Page view:
← Foundations of Quantum Theory →
Quantum theory is as much a theory of language as it is a theory of physics. Properties like wave function collapse apply to statements about what we know of a situation, not to physical reality. All that is required is a bit of mathematical trickery applied to a language describing general principles of measurement.
Quantum Logic
In mathematics, a
logic» is a formal language (from
logos, meaning "word" in greek). In a logic, statements, or propositions have a particular form and are given truth values. In classical,
Boolean», or two-valued logic, truth values are either
0 or
1, for statements which are
FALSE or
TRUE respectively.
Quantum logic» is a
many valued logic». In many valued logics, truth values between
0 and
1 are allowed for statements which are neither true nor false, but have some kind of intuitive level of truth somewhere in between.
Fuzzy logic» is an example of a many valued logic, much used in control engineering.
Probability theory» is also a many valued logic, described as
The Logic of Science», in the classic
book» by
E. T. Jaynes», and also called
Bayesian reasoning».
Quantum logic (due to
Birkhoff» and
von Neumann») is a language which tells us about possible results of measurements. It does not tell us, in any direct way, what happens between measurements (later, when we are fluent in the language, we can draw some conclusions). Quantum logic is usually treated in a very abstract, and somewhat obscure, way. This treatment aims keep it as concrete as possible and to show how to translate the mathematical symbols of standard quantum theory into statements in the English language.
Feynman» is famously quoted as having said “I think it is safe to say that no one understands quantum mechanics”. The plethora of “interpretations” seems to bear this out. Yet I claim quantum mechanics can be understood. In essence, what I will describe is the “orthodox”, or Dirac-Von Neumann interpretation. If I have contributed anything, it is to make the interpretation easier to understand, not to alter it in a fundamental way. When I read von Neumann on interpretation, I do not think that what he says is really different from what I say. However, von Neumann was one of the most brilliant mathematicians in history. He had a way of talking over one’s head, which is difficult to follow. Mathematics is a language, but the natural ability to treat it like an ordinary language is not really human. This goes beyond the more common ability to manipulate equations according to rules, which does not require understanding. One way to learn language is to start by translating simple phrases and sentences into English from a primer. This section is a primer in the language of quantum logic, as well as an introduction to the mathematical structure of quantum theory.
Quantum logic is often rejected as an interpretation of quantum mechanics on the ground that it consists of obscure truth values for simple propositions. In my view this is wrong. It is better described as intuitive truth values for sophisticated propositions. In order to understand it we need to understand what the propositions of quantum mechanics are, and how to translate them into ordinary language. We will understand that these propositions refer to hypothetical measurements, and are not, in general, strictly true or false, but have levels of truth somewhere in between. In this section I describe the language for measurements of position of a single interacting particle. Later sections will treat states of more than one particle, interactions between particles, and introduce the mysterious property of
spin».
Truth Values
Classical logic applies to sets of statements about the real world which are definitely true or definitely false. For example, when we make a statement,
P(x) = “The position of a particle is x”,
we tend to assume that it is definitely true or definitely false. Such statements are said to be sharp, meaning that they have truth values from the set
{0,1}. When it is the case that
P(x) is definitely either true or false then classical logic and classical mechanics apply.
We cannot say that a statement about the future is strictly true or false.
Q(x) = “When a measurement of position is done, the result will be x”.
We may, however, assign a probability to such a statement. If we consider probabilities as truth values, then probability theory is a many valued logic, applying to sentences in the future tense.
In quantum mechanics we also talk about situations in which there is not going to be a measurement. Hypothetical measurement results can be described using statements in the
subjunctive mood»:
R(x) = “If a measurement of position were done, then the result would be x”.
R(x) is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like
R(x), in the subjunctive mood.
Sentences Describing Hypothetical Measurement Results
Statements in the subjunctive consist of two clauses, the conditional clause
“If a measurement of position were done, …”, and the consequent clause
“…, then the result would be x”. Quantum mechanics is based on statements composed of a conditional clause and a consequent clause. To formally describe physics using mathematics, we need to be more precise. The conditional clause must also contain whatever information is known before measurement. This information comes from a prior measurement. We therefore discuss two measurements, the first to determine the condition and the second to determine the outcome, or consequence. We represent the results of these measurements symbolically.
The conditional clause, referring to the first measurement, is represented by a ket. It is described as a
formal conditional clause to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The basic conditional clauses, on which the language is built, refer to measurements of position:
RULE I. is the formal conditional clause “If measured position were x, …”.
<img title="A discrete lattice is chosen from which to define position kets" alt="
QuantumCovariance-1" src="images/quantumcovariance/
QuantumCovariance-1.gif"> Measured positions are always discrete values, determined by the range and resolution of a measurement apparatus. We are discussing hypothetical measurement, and in practice, rather than use the values of
x generated by a particular apparatus to generate the basic propositions (represented by blue dots), it is simpler to we use an equally spaced lattice, containing
N positions given by decimals terminating at some value beyond the best available resolution of any existing appearatus (cyan grid), where
N is a very large, but finite, number. An actual position found by a real apparatus is described by a set of points in the lattice (magenta set boundaries). To describe this we need to extend the language, by introducing a structure corresponding to
OR. In grammar
OR is a
coordinating conjunction». In logic and mathematics it is
logical disjunction».
OR will be represented by the symbol
+. The use of a discrete lattice has implications to the mathematical statement of
covariance in the quantum domain]]. This will be considered in
Quantum Covariance. For the immediate development, it will be assumed that
Lorentz covariance» holds in an approximation valid at the limit of experimental accuracy.
A simple logical disjunction does not distinguish the likelihood of the two possibilities;
“If measured position were x or y, …”. To express the idea that one possibility is more likely than the other we introduce a weighting. Thus, if the magnitude of
a is greater than that of
b, then
means
“if measured position were either x or y, but more likely x, …”. Precise values for
a and
b in a given situation will be determined as the language develops.
We want to be able to express many possibilities,
“if the particle were found at x or y or z or …”, and we want weighting between all options. This is done recursively, by starting from the basic conditional clauses and repeating the following rule as many times as we require:
RULE II. If and are formal conditional clauses, and a and b are complex numbers, then is a formal conditional clause.
The set of formal conditional clauses, or kets, now has the mathematical structure of an
N-dimensional
vector space. The symbol
H1(t) is used to denote this vector space, and consists of a family of statements we can make about the measurement of position of a single particle at time
t. The basic conditional clauses,
, described in rule I are a basis for
H1(t).
One may ask why we are using complex numbers to construct the propositions of quantum theory. So long as we are talking about measurements of position at a particular time, real numbers would work just as well. In fact, real numbers would be in some ways more precise, because complex numbers introduce an extra level of freedom. The reason for using them lies in what happens when we want to compare a measurement at one time with a measurement at another, for example, when we want to talk about motion. Before we can study that we need to complete the language for talking about measurements of position at a particular time. Kets are often known as
states. They are not actually states of a particle, but formal conditional clauses. That is something of a mouthful. I will often call them states, in keeping with common practice when no confusion arises.
A conditional clause on its own has little meaning. To complete a formal sentence in quantum theory we need to put it together with a consequent clause. Consequent causes refer to a second measurement, at the same time as the first measurement. For hypothetical measurements, there is no problem with the idea that they both take place at the same time. To make statements about real measurement results we will also need to know how kets change in time.
There is no fundamental difference between one measurement and another, so the grammatical structure,
weighted disjunction described in rule II, applies equally well to consequent clauses. These also form an
N-dimensional vector space defined from a basis of consequent clauses in one-one correspondence with the basic conditional clauses, or kets, described by rule I. Consequent clauses are represented symbolically by bras, according to rule III:
RULE III. is the formal consequent clause “…, then, in a second measurement at the same time, measured position would be x”.
We put the two clauses together, to make a braket, representing a statement about measurement at a given time:
RULE IV. is the statement “If measured position were y, then, in a second measurement at the same time, measured position would be x”.
From observation we know that, if, at some particular time, a particle is measured at position
x, then its position is definitely
x and it cannot be measured separately at some other position
y at the same time. The statement
is strictly true or false, depending on whether or not
x = y. Its truth value is given by a
Kronecker delta, and we write:
.
Taken together with
linearity and
complex conjugation, this is sufficient to define an
inner product between any two kets,
and
in
H1(t). Thus,
H1(t) is a
Hilbert space and the basic states of rule I are an
orthonormal basis.
Integral Representation
In the language of quantum theory, absolute magnitude has no meaning; only relative magnitudes are important in weighted logical
OR. For any complex number
a, the clause
means exactly the same thing as
. In other words,
normalisation is irrelevant. For large values of
N, it is often convenient to normalise basis states so that the inner product
becomes a
Dirac delta function instead of a Kronecker delta,
This normalisation is natural when using the language of
integrals rather than the language of sums. If we bear in mind that strictly the integrals stand for finite sums with
N terms, everything remains well defined, and we be able to control the divergence problems of
quantum electrodynamics». With this normalisation, the
resolution of unity,
is replaced by,
The integral is just what was defined previously, but uses three dimensions instead of one. It is regarded strictly as a sum of a
N terms where
N is a finite number of possible positions, rather than an infinite sum.
Plane Wave States
For a 3-vector,
p, at the origin, define a particular ket,
, as a sum of position states,
:
Taking the inner product with
defines a
sinusoidal wave,
This function can be seen to have planar wave crests, perpendicular to vector
p and at equally spaced intervals
2π/|p|. It is a
plane wave» at constant time.
Definition: is a plane wave state with momentum p.
Definition: The space of momenta, p, is momentum space.
Using this approach to quantum theory, this is the fundamental definition of 3-momentum. The justification for the definition is that it turns out later that
p is a conserved quantity which corresponds precisely to the classical notion of momentum. Momentum is here defined using
natural units», in which the
Dirac constant»,
. To convert to conventional units, substitute
Definition: For each ket, , define the momentum space wave function, f (p), such that
Using the dot product with the laws of indices, the
delta function in 3 dimensions is the product of three 1 dimensional delta functions. So,
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the
Fourier inversion theorem». In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over
p and over
x (or
y) really stand for finite sums with
N terms, where
N is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which
N goes to infinity, provided that
N is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in
Quantum Covariance and in
Discrete Quantum Electrodynamics.
The Resolution of Unity in Momentum Space
With a little rearrangement,
Using the
integral representation of the resolution of unity, substituting for
, and using the resolution of unity again, for any kets
and
,
So, we have another resolution of unity, this time in terms of momentum states:
So, any ket can be written as a sum of plane wave states,
This shows that plane wave states are a basis for Hilbert space. Mathematically, switching between representations in terms of momentum and position is simply a change of basis, analogous to a change of coordinate axes for 3-vectors.
Foundations of Quantum Theory ↑ Observable Quantities →
