Most recent edit on 2008-08-13 03:40:37 by CharlesFrancis

Additions:

←  Evolution of Quantum States  ↑  →

Deletions:

←  Evolution of Quantum States  →

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Edited on 2008-08-12 07:06:05 by CharlesFrancis

Additions:
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern Bayesian» ideas, probabilities» describe our ideas concerning the likelihood of events. They not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).

Deletions:
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability» Bayesian ideas, [[http://en.wikipedia.org/wiki/Probability» probabilities describe our ideas concerning the likelihood of events. They not a direct description of physical reality. Whether or not reality is fundamentally discrete, probabilities are properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).

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Edited on 2008-08-12 07:03:11 by CharlesFrancis

Additions:
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability» Bayesian ideas, [[http://en.wikipedia.org/wiki/Probability» probabilities describe our ideas concerning the likelihood of events. They not a direct description of physical reality. Whether or not reality is fundamentally discrete, probabilities are properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).

Deletions:
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. Probability theory» is a description of our ideas of likelihood», not a direct description of physical reality. Whether or not reality is fundamentally discrete, probabilities are properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).

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Edited on 2008-08-12 06:40:41 by CharlesFrancis

Additions:
If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. Probability theory» is a description of our ideas of likelihood», not a direct description of physical reality. Whether or not reality is fundamentally discrete, probabilities are properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).
At this point we have only states of a single, non-interacting particle. Subsequent pages will allow states containing an indefinite number of particles, and interactions betwen them. It is assumed that all phsysical systems will can be modeled in the same way. Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,
Ignoring terms in squares of dt, and using , ,

Deletions:
If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if
If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measurable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.
Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,
Ignoring terms in squares of dt, and using , ,

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Edited on 2008-08-07 04:25:32 by CharlesFrancis

Additions:
Since states can be chosen to be normalised we may require that U conserves the norm, i.e. for all ,

Deletions:
Since states can be chosen to be normalised we may require that U conserves the norm, i.e. for all t₀ is ,

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Edited on 2008-08-07 04:22:05 by CharlesFrancis

No differences.

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Edited on 2008-08-07 04:18:50 by CharlesFrancis

Additions:
Since states can be chosen to be normalised we may require that U conserves the norm, i.e. for all t₀ is ,

Applying this to ,

By linearity of U,

By linearity of the inner product,

Similarly,

Combining these results,
So U is unitary.

Deletions:
If the normalised state at time t₀ is then the probability for finding the normalised state in a measurement at time t₀ is

If, in the absence of outside influences, evolves to and and evolves to , then the probability that a measurement at time t₀ will give the result is unchanged,

To ensure that this is always so, we require

So, conservation of probability implies that U is unitary,

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Edited on 2008-05-14 23:06:39 by CharlesFrancis

Additions:
If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measurable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.
So, conservation of probability implies that U is unitary,

Deletions:
If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measureable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.
So, conservation of probability implies that U is unitary,

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Edited on 2008-02-29 03:22:30 by CharlesFrancis

Additions:
This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements. This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.

Deletions:
This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements.This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.

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Edited on 2008-02-29 03:22:02 by CharlesFrancis

Additions:
Proof: Differentiate the wave function using Stone’s theorem,

Deletions:
Proof: Differentiate the wave function using Stone’s theorem,

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Edited on 2008-02-29 03:12:58 by CharlesFrancis

Additions:
According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

Deletions:
According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

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Edited on 2008-02-29 02:27:38 by CharlesFrancis

Additions:
In words, U is a linear operator.
It follows from linearity of U that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,
Evolution of Quantum States ↑ The Dirac Equation →

Deletions:
In words, U is a linear operator. It follows that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,
Evolution of Quantum States↑ The Dirac Equation →

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Edited on 2008-01-15 06:41:03 by CharlesFrancis

Additions:

←  Evolution of Quantum States  →

Evolution of Quantum States↑ The Dirac Equation →

Deletions:

←  Evolution  →

Time Evolution ↑ The Dirac Equation →

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Edited on 2008-01-12 00:54:21 by CharlesFrancis

Additions:

←  Evolution  →

Deletions:

←  Time Evolution  →

Classical Behaviour

The evolution of a state is given by

Ehrenfest’s Theorem:

Proof: Differentiate using the product rule,

which establishes Ehrenfest’s theorem» and governs the classical behaviour of matter, given by the expectation of an observable.

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Edited on 2008-01-08 00:39:42 by CharlesFrancis

Additions:

←  Time Evolution  →

Time Evolution ↑ The Dirac Equation →

Deletions:

←  Time Evolution  →

Time Evolution ↑ States of Many Particles →

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Edited on 2008-01-04 02:58:59 by CharlesFrancis

Additions:

←  Time Evolution  →

The inner product allows us to calculate probabilities for the outcome of a measurement provided that we know the ket describing hypothetical measurement at the time of measurement. This is only useful if we can calculate the ket at any time, t, from a known previous measurement result. The probability interpretation requires that time evolution is determined from a first order wave equation, the Schrödinger equation. Relativistic considerations dictate that Newton’s first law is obeyed for non-interacting particles.

Linearity of Time Evolution

Hilbert space refers to measurement at time, t, so that , where a different Hilbert space is required at each time. Bold type will be used for 3-vectors. For the time being, it will be assumed that the scale on which quantum mechanics applies is such that curvature can be ignored, so that position coordinates can be denoted by displacement vectors in Minkowski spacetime. Position states at time x⁰ = t will be denoted .

Definition:  If at time t₀ the ket is in H¹(t₀), then the ket at time t is given by the time evolution operator, U(t, t₀) : H¹(t₀) → H¹(t₁), such that .
If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if

then

So,

In words, U is a linear operator. It follows that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,

Continuity of Time Evolution

If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measureable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.
Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,

does not depend on t₀. We require that the evolution in an interval t₁ + t₂ is the same as the evolution in t₁ followed by the evolution in t₂, and is also equal to the evolution in t₂ followed by the evolution in t₁.

In a zero time interval, there is no evolution. So, U(0) does not change the state.

Using a negative value of t reverses time evolution (put t = t₁ = −t₂).

Unitarity of Time Evolution

If the normalised state at time t₀ is then the probability for finding the normalised state in a measurement at time t₀ is

If, in the absence of outside influences, evolves to and and evolves to , then the probability that a measurement at time t₀ will give the result is unchanged,

To ensure that this is always so, we require

So, conservation of probability implies that U is unitary,

Stone’s Theorem

The derivative of U is

This prompts the definition of the Hamiltonian» operator, which does not depend on t.

Definition:  The Hamiltonian operator H : H¹(t)→H¹(t) is

We have

So

Since U is unitary, for a small time dt,

Ignoring terms in squares of dt, and using , ,

Using unitarity of U, we find that H is Hermitian, iH = H^†. We have the differential equation,

which has solution (as for a differential equation of a function)

This result was first proved by proved by Marshall Stone» in 1932, and is known as Stone’s theorem».

Classical Behaviour

The evolution of a state is given by

Ehrenfest’s Theorem:

Proof: Differentiate using the product rule,

which establishes Ehrenfest’s theorem» and governs the classical behaviour of matter, given by the expectation of an observable.

The Heisenberg Picture

I have formulated quantum theory in such a way that states evolve in time and observable operators are assumed to be constant. This is the Schrödinger picture». A precisely equivalent formulation, the Heisenberg picture» is found by a unitary transformation. States in the Heisenberg picture are defined by

and are constant. Then, an observable operator, A, in the Schrödinger picture, is given in the Heisenberg picture by

It is immediately clear that the result of calculation of probabilities is the same in both pictures,

Switching between the Schrödinger and Heisenberg pictures is simply a change of basis, and is precisely equivalent to coordinate transformation.

The Wave Function

On the assumption that the resolution of measurement may be arbitrarily fine, we define the wave function.

Definition:  The wave function is the map f : R⁴→C given by:

Theorem:  The wave function satisfies the Schrödinger Equation,

Proof: Differentiate the wave function using Stone’s theorem,

Newton’s First Law

According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

where E² = (p⁰)² = m² + p² for some constant m. Thus, p does not change in time,

establishing Newton’s first law. E is identified with energy and m with mass.

Definition:  The mass shell condition is the vector identity,

m² = E² − p².

Wave Mechanics

To find the evolution of a ket from an initial state, , at t = 0, calculate the momentum space wave function using the resolution of unity,

Then, using the resolution of unity in momentum space,

By linearity of the time evolution operator, U,

The coefficients are

Thus, the momentum space wave function is constant in time. If we know the ket from a measurement at time t = 0, we can calculate the ket for a measurement at any other time, and hence probabilities for the results of measurement at any time, by the usual methods of wave mechanics.
This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements.This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.
Time Evolution ↑ States of Many Particles →

Deletions:

←  Observable Quantities  →

It has been said that measurement of time and position sufficient for the study of all physical quantities. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bubble chamber, being a set of positions over a time interval. In the most general case, measurement of position might refer to the position of particles other than the one under study, such as the position of a pointer. The language of Quantum logic is here found sufficient to discuss probabilities for the results of any measurement where this is so.

Probability Interpretation

To make the language precise, we need to attach numerical values to the complex numbers introduced in rule II. To determine a complex number we must determine its magnitude and phase. Phase contains information on the evolution of kets, and is found from relativistic considerations. Magnitude will be determined from probability. It only makes sense to talk about probability when we are actually going to do a measurement. When we are actually going to do the measurement, a statement about hypothetical measurement, in the subjunctive mood, automatically becomes a statement about real measurement, in the future tense. This being the case, truth values for hypothetical results must become truth values for future events, i.e. probabilities. We require that the language generates the probability, , of getting the result, x, when a measurement of position is actually going to be performed on the state .
Two conditions are required of a probability density function». It must be greater than or equal to zero for all values of its argument, and it must have an area under its graph equal to 1 — or, in 3 dimensions, a hyper-volume under its graph of 1. Using the resolution of unity,

So, for a positive definite inner product, the function,

has the property of a probability density. Using the resolution of unity, we write as a sum of the basis states of exact position,

Definition:  The magnitudes of the coefficients, , are such that, for all x,

is the conditional probability that a measurement of position will return the result x, given the information from a prior measurement contained in the ket .
This definition ties down the relative magnitudes of the complex numbers, a and b, in rule II, and makes precise the weighting in logical OR.

Measurement

Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.

Definition:  A measurement of a physical quantity is any physical process such that a determination of the quantity is possible in principle.
This is a very broad definition of measurement, and does not require the intervention of a human observer. In classical physics, complete knowledge of the situation at a particular time contains sufficient information to determine the situation at any other time. When classical laws apply, and the state at some time is known from measurement, the state at all times can be calculated, at least in principle. For example, we can reasonably assume that Venus continues on its orbit while it is on the far side of the Sun and cannot be observed. Experience tells us that when Venus reappears it will be observed in precisely the position found by calculation from prior measurements. It is unreasonable to think that the same calculations do not hold while Venus is not observed. When classical laws apply, and the state has been measured at some time, then the state can be said to be effectively measured at any time, whether or not it is actually observed, and whether or not a calculation is actually performed.

Quantum Measurement

Measurement has two effects on the state of a particle, altering it due to the interaction of the apparatus with the particle, and also changing the information we have about the state. New information causes a change of state even in the absence of physical change because the state, or ket, is just a label for available information. The ket changes in part due to the effect of the apparatus on the particle, and in part due to the change in a conditional probability when the condition becomes known. This inverts the measurement problem»; collapse represents a change in information due to a new measurement but Schrödinger's equation» requires explanation — interference patterns are real. The requirement for a wave equation will be found in the next section, Time Evolution.
Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in nature.
When we do a measurement, K, we get a definite result, a terminating decimal or n-tuple of terminating decimals read off the measurement apparatus. Let the possible results be k_i for i = 1, …, n . We assume that the dimension of H¹ is greater than n; this must be so if all measurements are reducible to measurements of position, and can be ensured through an appropriate choice of lattice at finer than the resolution of measurement. Each physical state is associated with a ket, labelled by the measurement result, so that if the measured result is k_i then the state is . The empirical determination of as a member of H¹ requires that we draw from experimental data the value of the inner product for an arbitrary state, . Without loss of generality and are normalised. By assumption, measurement of K is reducible to a set of measurements of position, so that each k_i is in one to one correspondence with the positions y_i of one or more particles used for the measurement (e.g. y_i may be the positions of one or more pointers). Then,

is the probability that a measurement of K has result k_i, given the initial state in H¹.
If the result is k_i, it is definitely k_i and cannot be given as k_j with i ≠ j by another measurement of K at the same time. So,

Measurement with result, k_i, implies a physical action on a system and is represented by the action of an operator, K_i, on Hilbert space. If a quantity is measurable we require that there is an element of physical reality associated with its measurement, meaning that, in measurement, the configuration of particles necessarily becomes such that the quantity has a well defined value. In practice, this means that, in the limit in which the time between two measurements goes to zero, a second measurement of the quantity necessarily gives the same result as the first. It follows that K_i is a projection operator», and has the form

.

Definition:  A projection operator, K, is a linear operator such that K² = K.

Definition: The projection postulate states that the effect of a measurement is equivalent to the action of a projection operator.

Observable Operators

The expectation» of the result from a measurement of K, given an initial normalised state, in H¹, is

Definition:  An observable^» is a Hermitian operator with the form,

The effect of K on the state is to leave it unchanged.

k_i is the value of K in the state .

Definition:  If, for some observable, K, for some number, k, and for some state , , then k is an eigenvalue and is an eigenstate of K.
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue».
In a measurement of K, the probability that operators describing the interactions combine to K_i is

.

Then can be understood as a classical probability function, where the random variable runs over the set of projection operators, K_i, corresponding to the outcomes of the measurement. The physical interpretation is that each K_i represents a set of unknown configurations of particle interactions in measurement, namely that set of configurations leading to the result k_i.
More generally, we have that if the state at time t is known to be , then the probability that a measurement at time t will give the result is given by

Definition:  To complete the structure of a many valued logic, we regard

as the truth value of the proposition .

The Classical Correspondence

In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. A classical property is the expectation of the corresponding observable in the limit of large sample behaviour, as (not as sometimes stated; Planck’s constant is simply a change of scale from natural to conventional units and it would be meaningless to let it go to zero). A precise treatment of classical quantities requires the development of the interacting theory. It will be shown that determinate laws obtain for classical quantities. For now, we will simply assume determinate laws for expectations in the large number limit. For example, the centre of gravity of a macroscopic body is a weighted average of the positions of the elementary particles which constitute it. Schrödinger’s cat is definitely either alive or dead because, consisting as it does of a large number of elementary particles, its properties are expectations obeying classical laws. The state, or ket, simply encodes probability and the cat may be described as a superposition until the box is opened.

The Uncertainty Principle

The uncertainty principle» can be derived from the theory while the theory cannot be derived from it. Strictly, this makes it a theorem, not a principle. Uncertainty in an observable is defined in the same way as variance» in statistics. It is a prediction of the root mean squared difference between observed values and the expected value».

Definition:  The uncertainty, ΔK in an observable operator, K, is

Definition:  The commutator between operators A and B is [A, B] = AB − BA.

Theorem:  Any two observables A and B obey the general uncertainty relation,

Proof:  Let and . Then, since A and B are Hermitian,

by the Cauchy-Schwarz inequality. Thus,

The general uncertainty relation follows by applying this result to and , observing that .
The general uncertainty relation can be applied to any two observables. For example, the position observable is

and the momentum observable is

as may be seen by differentiating the coefficients for plane wave states. Then we find, by applying the product rule for differentiation, for uncertainty in position-momentum,

This gives the uncertainty relation in natural units. In conventional units we have

This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927. We read the uncertainty relation for time-energy from the time components, E = p⁰ and t = x⁰,

Observables ↑ Time Evolution →

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Edited on 2008-01-04 02:58:13 by CharlesFrancis

Additions:

←  Observable Quantities  →

It has been said that measurement of time and position sufficient for the study of all physical quantities. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance, and a typical measurement of momentum of a particle involves plotting its path in a bubble chamber, being a set of positions over a time interval. In the most general case, measurement of position might refer to the position of particles other than the one under study, such as the position of a pointer. The language of Quantum logic is here found sufficient to discuss probabilities for the results of any measurement where this is so.

Probability Interpretation

To make the language precise, we need to attach numerical values to the complex numbers introduced in rule II. To determine a complex number we must determine its magnitude and phase. Phase contains information on the evolution of kets, and is found from relativistic considerations. Magnitude will be determined from probability. It only makes sense to talk about probability when we are actually going to do a measurement. When we are actually going to do the measurement, a statement about hypothetical measurement, in the subjunctive mood, automatically becomes a statement about real measurement, in the future tense. This being the case, truth values for hypothetical results must become truth values for future events, i.e. probabilities. We require that the language generates the probability, , of getting the result, x, when a measurement of position is actually going to be performed on the state .
Two conditions are required of a probability density function». It must be greater than or equal to zero for all values of its argument, and it must have an area under its graph equal to 1 — or, in 3 dimensions, a hyper-volume under its graph of 1. Using the resolution of unity,

So, for a positive definite inner product, the function,

has the property of a probability density. Using the resolution of unity, we write as a sum of the basis states of exact position,

Definition:  The magnitudes of the coefficients, , are such that, for all x,

is the conditional probability that a measurement of position will return the result x, given the information from a prior measurement contained in the ket .
This definition ties down the relative magnitudes of the complex numbers, a and b, in rule II, and makes precise the weighting in logical OR.

Measurement

Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.

Definition:  A measurement of a physical quantity is any physical process such that a determination of the quantity is possible in principle.
This is a very broad definition of measurement, and does not require the intervention of a human observer. In classical physics, complete knowledge of the situation at a particular time contains sufficient information to determine the situation at any other time. When classical laws apply, and the state at some time is known from measurement, the state at all times can be calculated, at least in principle. For example, we can reasonably assume that Venus continues on its orbit while it is on the far side of the Sun and cannot be observed. Experience tells us that when Venus reappears it will be observed in precisely the position found by calculation from prior measurements. It is unreasonable to think that the same calculations do not hold while Venus is not observed. When classical laws apply, and the state has been measured at some time, then the state can be said to be effectively measured at any time, whether or not it is actually observed, and whether or not a calculation is actually performed.

Quantum Measurement

Measurement has two effects on the state of a particle, altering it due to the interaction of the apparatus with the particle, and also changing the information we have about the state. New information causes a change of state even in the absence of physical change because the state, or ket, is just a label for available information. The ket changes in part due to the effect of the apparatus on the particle, and in part due to the change in a conditional probability when the condition becomes known. This inverts the measurement problem»; collapse represents a change in information due to a new measurement but Schrödinger's equation» requires explanation — interference patterns are real. The requirement for a wave equation will be found in the next section, Time Evolution.
Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in nature.
When we do a measurement, K, we get a definite result, a terminating decimal or n-tuple of terminating decimals read off the measurement apparatus. Let the possible results be k_i for i = 1, …, n . We assume that the dimension of H¹ is greater than n; this must be so if all measurements are reducible to measurements of position, and can be ensured through an appropriate choice of lattice at finer than the resolution of measurement. Each physical state is associated with a ket, labelled by the measurement result, so that if the measured result is k_i then the state is . The empirical determination of as a member of H¹ requires that we draw from experimental data the value of the inner product for an arbitrary state, . Without loss of generality and are normalised. By assumption, measurement of K is reducible to a set of measurements of position, so that each k_i is in one to one correspondence with the positions y_i of one or more particles used for the measurement (e.g. y_i may be the positions of one or more pointers). Then,

is the probability that a measurement of K has result k_i, given the initial state in H¹.
If the result is k_i, it is definitely k_i and cannot be given as k_j with i ≠ j by another measurement of K at the same time. So,

Measurement with result, k_i, implies a physical action on a system and is represented by the action of an operator, K_i, on Hilbert space. If a quantity is measurable we require that there is an element of physical reality associated with its measurement, meaning that, in measurement, the configuration of particles necessarily becomes such that the quantity has a well defined value. In practice, this means that, in the limit in which the time between two measurements goes to zero, a second measurement of the quantity necessarily gives the same result as the first. It follows that K_i is a projection operator», and has the form

.

Definition:  A projection operator, K, is a linear operator such that K² = K.

Definition: The projection postulate states that the effect of a measurement is equivalent to the action of a projection operator.

Observable Operators

The expectation» of the result from a measurement of K, given an initial normalised state, in H¹, is

Definition:  An observable^» is a Hermitian operator with the form,

The effect of K on the state is to leave it unchanged.

k_i is the value of K in the state .

Definition:  If, for some observable, K, for some number, k, and for some state , , then k is an eigenvalue and is an eigenstate of K.
If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue».
In a measurement of K, the probability that operators describing the interactions combine to K_i is

.

Then can be understood as a classical probability function, where the random variable runs over the set of projection operators, K_i, corresponding to the outcomes of the measurement. The physical interpretation is that each K_i represents a set of unknown configurations of particle interactions in measurement, namely that set of configurations leading to the result k_i.
More generally, we have that if the state at time t is known to be , then the probability that a measurement at time t will give the result is given by

Definition:  To complete the structure of a many valued logic, we regard

as the truth value of the proposition .

The Classical Correspondence

In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. A classical property is the expectation of the corresponding observable in the limit of large sample behaviour, as (not as sometimes stated; Planck’s constant is simply a change of scale from natural to conventional units and it would be meaningless to let it go to zero). A precise treatment of classical quantities requires the development of the interacting theory. It will be shown that determinate laws obtain for classical quantities. For now, we will simply assume determinate laws for expectations in the large number limit. For example, the centre of gravity of a macroscopic body is a weighted average of the positions of the elementary particles which constitute it. Schrödinger’s cat is definitely either alive or dead because, consisting as it does of a large number of elementary particles, its properties are expectations obeying classical laws. The state, or ket, simply encodes probability and the cat may be described as a superposition until the box is opened.

The Uncertainty Principle

The uncertainty principle» can be derived from the theory while the theory cannot be derived from it. Strictly, this makes it a theorem, not a principle. Uncertainty in an observable is defined in the same way as variance» in statistics. It is a prediction of the root mean squared difference between observed values and the expected value».

Definition:  The uncertainty, ΔK in an observable operator, K, is

Definition:  The commutator between operators A and B is [A, B] = AB − BA.

Theorem:  Any two observables A and B obey the general uncertainty relation,

Proof:  Let and . Then, since A and B are Hermitian,

by the Cauchy-Schwarz inequality. Thus,

The general uncertainty relation follows by applying this result to and , observing that .
The general uncertainty relation can be applied to any two observables. For example, the position observable is

and the momentum observable is

as may be seen by differentiating the coefficients for plane wave states. Then we find, by applying the product rule for differentiation, for uncertainty in position-momentum,

This gives the uncertainty relation in natural units. In conventional units we have

This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927. We read the uncertainty relation for time-energy from the time components, E = p⁰ and t = x⁰,

Observables ↑ Time Evolution →

Deletions:

←  Time Evolution  →

The inner product allows us to calculate probabilities for the outcome of a measurement provided that we know the ket describing hypothetical measurement at the time of measurement. This is only useful if we can calculate the ket at any time, t, from a known previous measurement result. The probability interpretation requires that time evolution is determined from a first order wave equation, the Schrödinger equation. Relativistic considerations dictate that Newton’s first law is obeyed for non-interacting particles.

Linearity of Time Evolution

Hilbert space refers to measurement at time, t, so that , where a different Hilbert space is required at each time. Bold type will be used for 3-vectors. For the time being, it will be assumed that the scale on which quantum mechanics applies is such that curvature can be ignored, so that position coordinates can be denoted by displacement vectors in Minkowski spacetime. Position states at time x⁰ = t will be denoted .

Definition:  If at time t₀ the ket is in H¹(t₀), then the ket at time t is given by the time evolution operator, U(t, t₀) : H¹(t₀) → H¹(t₁), such that .
If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if

then

So,

In words, U is a linear operator. It follows that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,

Continuity of Time Evolution

If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measureable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.
Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,

does not depend on t₀. We require that the evolution in an interval t₁ + t₂ is the same as the evolution in t₁ followed by the evolution in t₂, and is also equal to the evolution in t₂ followed by the evolution in t₁.

In a zero time interval, there is no evolution. So, U(0) does not change the state.

Using a negative value of t reverses time evolution (put t = t₁ = −t₂).

Unitarity of Time Evolution

If the normalised state at time t₀ is then the probability for finding the normalised state in a measurement at time t₀ is

If, in the absence of outside influences, evolves to and and evolves to , then the probability that a measurement at time t₀ will give the result is unchanged,

To ensure that this is always so, we require

So, conservation of probability implies that U is unitary,

Stone’s Theorem

The derivative of U is

This prompts the definition of the Hamiltonian» operator, which does not depend on t.

Definition:  The Hamiltonian operator H : H¹(t)→H¹(t) is

We have

So

Since U is unitary, for a small time dt,

Ignoring terms in squares of dt, and using , ,

Using unitarity of U, we find that H is Hermitian, iH = H^†. We have the differential equation,

which has solution (as for a differential equation of a function)

This result was first proved by proved by Marshall Stone» in 1932, and is known as Stone’s theorem».

Classical Behaviour

The evolution of a state is given by

Ehrenfest’s Theorem:

Proof: Differentiate using the product rule,

which establishes Ehrenfest’s theorem» and governs the classical behaviour of matter, given by the expectation of an observable.

The Heisenberg Picture

I have formulated quantum theory in such a way that states evolve in time and observable operators are assumed to be constant. This is the Schrödinger picture». A precisely equivalent formulation, the Heisenberg picture» is found by a unitary transformation. States in the Heisenberg picture are defined by

and are constant. Then, an observable operator, A, in the Schrödinger picture, is given in the Heisenberg picture by

It is immediately clear that the result of calculation of probabilities is the same in both pictures,

Switching between the Schrödinger and Heisenberg pictures is simply a change of basis, and is precisely equivalent to coordinate transformation.

The Wave Function

On the assumption that the resolution of measurement may be arbitrarily fine, we define the wave function.

Definition:  The wave function is the map f : R⁴→C given by:

Theorem:  The wave function satisfies the Schrödinger Equation,

Proof: Differentiate the wave function using Stone’s theorem,

Newton’s First Law

According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

where E² = (p⁰)² = m² + p² for some constant m. Thus, p does not change in time,

establishing Newton’s first law. E is identified with energy and m with mass.

Definition:  The mass shell condition is the vector identity,

m² = E² − p².

Wave Mechanics

To find the evolution of a ket from an initial state, , at t = 0, calculate the momentum space wave function using the resolution of unity,

Then, using the resolution of unity in momentum space,

By linearity of the time evolution operator, U,

The coefficients are

Thus, the momentum space wave function is constant in time. If we know the ket from a measurement at time t = 0, we can calculate the ket for a measurement at any other time, and hence probabilities for the results of measurement at any time, by the usual methods of wave mechanics.
This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements.This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.
Time Evolution ↑ States of Many Particles →

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Edited on 2007-12-29 23:02:35 by CharlesFrancis

Additions:
where E² = (p⁰)² = m² + p² for some constant m. Thus, p does not change in time,

Deletions:
where E² = (p⁰)² = m² + p² for some constant m. Thus p does not change in time,

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Edited on 2007-12-29 22:59:41 by CharlesFrancis

Additions:
According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

Deletions:
According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations.
In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,

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Oldest known version of this page was edited on 2007-12-29 22:58:03 by CharlesFrancis []

←  Time Evolution  →

Linearity of Time Evolution

Hilbert space refers to measurement at time, t, so that , where a different Hilbert space is required at each time. Bold type will be used for 3-vectors. For the time being, it will be assumed that the scale on which quantum mechanics applies is such that curvature can be ignored, so that position coordinates can be denoted by displacement vectors in Minkowski spacetime. Position states at time x⁰ = t will be denoted .


If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if
then
So,
In words, U is a linear operator. It follows that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,

Continuity of Time Evolution

If the fundamental building blocks of matter are particles, then one would expect that interactions between particles are discrete. If this is so, then time evolution cannot be precisely modelled by a continuous equation. Nonetheless, the proper time between discrete interactions may be expected to be extremely small, and certainly much smaller than can be directly measured. In this case it will be reasonable to approximate time evolution with a continuous operator, U. In the absence of interactions, we do not expect a measureable difference between a discrete treatment and a continuous treatment. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted). In practice, the issues concerning continuity are deep and far reaching, and it will be necessary to consider them again when interactions are considered.

Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,
does not depend on t₀. We require that the evolution in an interval t₁ + t₂ is the same as the evolution in t₁ followed by the evolution in t₂, and is also equal to the evolution in t₂ followed by the evolution in t₁.
In a zero time interval, there is no evolution. So, U(0) does not change the state.
Using a negative value of t reverses time evolution (put t = t₁ = −t₂).

Unitarity of Time Evolution

If the normalised state at time t₀ is then the probability for finding the normalised state in a measurement at time t₀ is
If, in the absence of outside influences, evolves to and and evolves to , then the probability that a measurement at time t₀ will give the result is unchanged,
To ensure that this is always so, we require
So, conservation of probability implies that U is unitary,

Stone’s Theorem

The derivative of U is
This prompts the definition of the Hamiltonian» operator, which does not depend on t.


We have
So
Since U is unitary, for a small time dt,
Ignoring terms in squares of dt, and using , ,
Using unitarity of U, we find that H is Hermitian, iH = H^†. We have the differential equation,
which has solution (as for a differential equation of a function)
This result was first proved by proved by Marshall Stone» in 1932, and is known as Stone’s theorem».

Classical Behaviour

The evolution of a state is given by


Proof: Differentiate using the product rule,
which establishes Ehrenfest’s theorem» and governs the classical behaviour of matter, given by the expectation of an observable.

The Heisenberg Picture

I have formulated quantum theory in such a way that states evolve in time and observable operators are assumed to be constant. This is the Schrödinger picture». A precisely equivalent formulation, the Heisenberg picture» is found by a unitary transformation. States in the Heisenberg picture are defined by
and are constant. Then, an observable operator, A, in the Schrödinger picture, is given in the Heisenberg picture by
It is immediately clear that the result of calculation of probabilities is the same in both pictures,
Switching between the Schrödinger and Heisenberg pictures is simply a change of basis, and is precisely equivalent to coordinate transformation.

The Wave Function

On the assumption that the resolution of measurement may be arbitrarily fine, we define the wave function.



Proof: Differentiate the wave function using Stone’s theorem,

Newton’s First Law

According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations.
In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave» evolves in the usual manner,
where E² = (p⁰)² = m² + p² for some constant m. Thus p does not change in time,
establishing Newton’s first law. E is identified with energy and m with mass.

Wave Mechanics

To find the evolution of a ket from an initial state, , at t = 0, calculate the momentum space wave function using the resolution of unity,
Then, using the resolution of unity in momentum space,
By linearity of the time evolution operator, U,
The coefficients are
Thus, the momentum space wave function is constant in time. If we know the ket from a measurement at time t = 0, we can calculate the ket for a measurement at any other time, and hence probabilities for the results of measurement at any time, by the usual methods of wave mechanics.

This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements.This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.

Time Evolution ↑ States of Many Particles →