Most recent edit on 2008-08-13 03:43:52 by CharlesFrancis
Additions:
← Particle Interactions ↑ →
Interactions are modelled as a perturbation to the motion of free particles, using quantum logical
OR to write the statement that, at each instant, either a particle interacts with another particle, or it does not, in which case its wave function evolves as a free particle. Relativistic considerations are used to derive the locality condition, showing that particles must meet in order to interact and which gives meaning to the claim that particles are point-like. Conservation of 3-momentum is demonstrated, showing that classical Newtonian mechanics is a consequence of the relational principles described in relativity and quantum logic.
We assume that we can define a Hermitian interaction density operator, I(x), such that, if we were to determine the time and position at which the interaction takes place, the probability that it takes place at x, is 
. The general principle of relativity implies that I(x) has equal effect on a matter anywhere and at any time. So, by the identification of addition with quantum logical , HI(t) can be written as a sum (as usual on this site, an integral represents a finite sum with enough terms that taking more makes no practical difference to calculation).
The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent OR between possibities. In this interpretation, I(x) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s path integral», or “sum over all paths” has as natural interpretation, not that a particle passes through all paths in spacetime (as described by Feynman, e.g. in QED: The Strange Theory of Light and Matter», but that the sum over paths is a logical OR between the possible paths that might be detected if an experiment could be done to trace the path. The time-ordered graphs give a pictorial representation of the same statement.
The locality condition also gives meaning to the statement that particles are point-like objects, since it shows that showing that two particles must meet at a point in order to interact. The meaning of the integral over space in the interaction Hamiltonian is logical disjunction, OR, meaning that a measurement to determine the position of the interaction could find any position, but it would always find the particles interacting at a point.
Deletions:
← Particle Interactions →
Interactions are modelled as a perturbation to the motion of free particles, using quantum logical OR to write the statement that, at each instant, either a particle interacts with another particle, or it does not, in which case its wave function evolves as a free particle. Relativistic considerations are used to derive the locality condition, showing that particles must meet in order to interact and which gives meaning to the claim that particles are point-like. Conservation of 3-momentum is demonstrated, showing that classical Newtonian mechanics is a consequence of the relational principles described in relativity and quantum logic.
In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, time evolution including the possibility of an interaction is described by a Hamiltonian, H, with
We assume that we can define a Hermitian interaction density operator, I(x), such that, if we were to determine the time and position at which the interaction takes place, the probability that it takes place at x, is . The general principle of relativity implies that I(x) has equal effect on a matter anywhere and at any time. So, by the identification of addition with quantum logical OR, HI(t) can be written as a sum (as usual on this site, an integral represents a finite sum with enough terms that taking more makes no practical difference to calculation).
The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent OR between possibities. In this interpretation, I(x) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s path integral», or “sum over all paths” has as natural interpretation, not that a particle passes through all paths in spacetime (as described by Feynman, e.g. in QED: The Strange Theory of Light and Matter», but that the sum over paths is a logical OR between the possible paths that might be detected if an experiment could be done to trace the path. The time-ordered graphs give a pictorial representation of the same statement.
The locality condition also gives meaning to the statement that particles are point-like objects, since it shows that showing that two particles must meet at a point in order to interact. The meaning of the integral over space in the interaction Hamiltonian is logical disjunction, OR, meaning that a measurement to determine the position of the interaction could find any position, but it would always find the particles interacting at a point.
Edited on 2008-06-28 09:25:42 by CharlesFrancis
Additions:
Definition: The Hamiltonian density or interaction density, I(x), is a Hermitian operator such that the interaction Hamiltonian is
Deletions:
Definition: The Hamiltonian density or interaction density is a Hermitian operator such that the interaction Hamiltonian is
Edited on 2008-05-05 01:13:07 by CharlesFrancis
Additions:
In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are found from the expectations of the corresponding observables 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). 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, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.
Ehrenfest’s theorem describes change in the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measurable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.
 I(x) can be represented diagrammatically as a vertex or node. The lines above the node correspond to creation operators, and those below the node correspond to annihilation operators. |
The perturbation expansion for 
generates a braket between each annihilation operator, 
, and every earlier creation operator, 
, and every particle in 
, and a braket between every creation operator, 
, and every particle in the final state, 
, All other brakets are zero. These brakets can be represented graphically by connecting corresponding vertices. Lines representing particles are shown with arrows from bottom to top, and lines representing antiparticles with arrows from top to bottom. Then the nth term of the perturbation expansion is a sum of terms, each represented as a time-ordered graph» containing n vertices.
Deletions:
In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are found from the expectation of the corresponding observables 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). 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, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.
Ehrenfest’s theorem describes change in the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measureable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.
 I(x) can be represented diagrammatically as a vertex or node. The lines above the node correspond to creation operators, and those below the node correspond to annihilation operators. |
The perturbation expansion for generates a braket between each annihilation operator, , and every earlier creation operator, , and every particle in , and a braket between every creation operator, , and every particle in the final state, , All other brakets are zero. These brakets can be represented graphically by connecting corresponding vertices. Lines representing particles are shown with arrows from bottom to top, and lines representing antiparticles with arrows from top to bottom. Then the nth term of the perturbation expansion is a sum of terms, each represented as a time-ordered graph» containing n vertices.
Edited on 2008-03-11 10:57:21 by CharlesFrancis
Additions:
← Particle Interactions →
Particle Interactions ↑ Quantum Electrodynamics →
Deletions:
← Particle Interactions →
Particle Interactions ↑ Quantum Electrodynamics →
Edited on 2008-03-09 04:18:09 by CharlesFrancis
Additions:
The Hamiltonian Density
Deletions:
The Hamiltonian Density
Edited on 2008-03-07 02:30:00 by CharlesFrancis
Additions:
So, the full Hamiltonian in the interaction picture is H0 + HI.
Deletions:
So, the full Hamiltonian in the interactioni picture is H0 + HI.
Edited on 2008-03-05 01:18:23 by CharlesFrancis
Additions:
Proof: We have (from above), for any 
,
Deletions:
Proof: We have, for any ,
Edited on 2008-03-03 02:45:42 by CharlesFrancis
Additions:
Proof: Since space translation is the same for an observable operator, A(x), and the corresponding expectation, 
we have
The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent OR between possibities. In this interpretation, I(x) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s path integral», or “sum over all paths” has as natural interpretation, not that a particle passes through all paths in spacetime (as described by Feynman, e.g. in QED: The Strange Theory of Light and Matter», but that the sum over paths is a logical OR between the possible paths that might be detected if an experiment could be done to trace the path. The time-ordered graphs give a pictorial representation of the same statement.
Deletions:
Proof: Since space translation is the same for an observable operator, A(x), and the corresponding classical observable, we have
The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent OR between possibities. In this interpretation, I(x) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s path integral», or “sum over all paths” has as natural interpretation, not that a particle passes through all paths in space-time (as described by Feynman, e.g. in QED: The Strange Theory of Light and Matter», but that the sum over paths is a logical OR between the possible paths that might be detected if an experiment could be done to trace the path. The time-ordered graphs give a pictorial representation of the same statement.
Edited on 2008-03-03 00:32:45 by CharlesFrancis
Additions:
which establishes Ehrenfest’s theorem». In particular, for an observable quantity with no explicit time dependence,
To give Ehrenfest’s theorem a relativistic form, we also need to describe the effect of change in position.
Ehrenfest’s theorem describes change in the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measureable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.
In the interaction picture, Ehrenfest’s theorem states that the evolution of the expectation of an observable quantity which is not specifically time dependent (i.e. whose value depends only on the configuration of matter) is given by the commutator of the observable with the interaction Hamiltonian (repeat the proof in the interaction picture). Observable quantities are the result of physical measurement processes, which depend on the interactions of matter, so the commutator is zero outside the light cone. It follows that no observable effects may be transmitted faster than the speed of light. Of course, this argument is circular, since it depends on the premise of special relativity that information may not travel faster than the speed of light, but it serves to illustrate the physical meaning of the locality condition.
Deletions:
which establishes Ehrenfest’s theorem» and governs the classical behaviour of matter. In particular, for an observable quantity with no explicit time dependence,
Ehrenfest’s theorem describes the time evolution of the expectation. To give it a relativistic form, we also need to describe the effect of change in position.
Ehrenfest’s theorem describes the time evolution of the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measureable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.
In the interaction picture, Ehrenfest’s theorem states that the evolution of a classical observable quantity which is not specifically time dependent (i.e. whose value depends only on the configuration of matter) is given by the commutator of the observable with the interaction Hamiltonian (repeat the proof in the interaction picture). Observable quantities are the result of physical measurement processes, which depend on the interactions of matter, so the commutator is zero outside the light cone. It follows that no observable effects may be transmitted faster than the speed of light. Of course, this argument is circular, since it depends on the premise of special relativity that information may not travel faster than the speed of light, but it serves to illustrate the physical meaning of the locality condition.
Edited on 2008-03-03 00:23:33 by CharlesFrancis
No differences.
Edited on 2008-03-03 00:21:04 by CharlesFrancis
Additions:
Ehrenfest’s theorem describes the time evolution of the expectation. To give it a relativistic form, we also need to describe the effect of change in position.
Edited on 2008-03-03 00:14:08 by CharlesFrancis
Additions:
In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are found from the expectation of the corresponding observables 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). 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, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.
Ehrenfest’s theorem describes the time evolution of the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measureable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.
Deletions:
In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are given by the expectation of the corresponding observables 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). 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, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.
Ehrenfest’s theorem describes the time evolution of a classical observable, A, provided that measurement of the observable does not alter it’s evolution (it applies to momentum, but not to measurement of position).
Edited on 2008-02-29 04:33:52 by CharlesFrancis
Additions:
Definition: The free Hamiltonian, H0, describes the evolution of kets for non-interacting particles.
Definition: The interaction Hamiltonian, Hint, describes the change in kets due to an interaction between particles.
Deletions:
Definition: The free Hamiltonian, H0 describes the evolution of kets for non-interacting particles.
Definition: The interaction Hamiltonian, Hint describes the change in kets due to an interaction between particles.
Edited on 2008-02-29 04:32:26 by CharlesFrancis
Additions:
In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, time evolution including the possibility of an interaction is described by a Hamiltonian, H, with
where H0 is the free Hamiltonian, and Hint is a Hermitian operator describing an interaction between particles, called the interaction Hamiltonian.
Definition: The free Hamiltonian, H0 describes the evolution of kets for non-interacting particles.
Definition: The interaction Hamiltonian, Hint describes the change in kets due to an interaction between particles.
In general, Hint will be a sum of terms for different types of interaction. For simplicity, I will consider only one type of interaction and spin indices will be suppressed. Spin is necessary for the full treatment of a specific interaction Hamiltonian, but has no bearing on the material on this page. Thus, the evolution of a state is given by
So, the full Hamiltonian in the interactioni picture is H0 + HI.
Deletions:
In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, the possibility of an interaction is described by a Hamiltonian, H, with
where free Hamiltonian, H0, is the Hamiltonian for non-interacting particles, and the interaction Hamiltonian, Hint, is a Hermitian operator describing the possibility of interaction. In general, Hint will be a sum of terms for different types of interaction. For simplicity, I will consider only one type of interaction and spin indices will be suppressed. Spin is necessary for the full treatment of a specific interaction Hamiltonian, but has no bearing on the material on this page. Thus, the evolution of a state is given by
So, the full Hamiltonian is H0 + HI.
Edited on 2008-02-29 04:07:14 by CharlesFrancis
Additions:
Ehrenfest’s theorem describes the time evolution of a classical observable, A, provided that measurement of the observable does not alter it’s evolution (it applies to momentum, but not to measurement of position).
Deletions:
Ehrenfest’s theorem describes the time evolution of a classical observable, A .
Edited on 2008-02-29 03:59:48 by CharlesFrancis
Additions:
In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, the possibility of an interaction is described by a Hamiltonian, H, with
Deletions:
In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, the possibility of an interaction is described by a Hamiltonian, H, with
Edited on 2008-02-29 03:16:18 by CharlesFrancis
Additions:
to observables. This is the interaction picture» In the absence of interaction, this would be the Heisenberg picture, and states would be constant in time. The interaction picture is useful for studying perturbations relative to to the behaviour of non-interacting particles, i.e. inertial matter, and implies that we are working in an inertial reference frame.
Deletions:
to observables. This is the interaction picture» In the absence of interaction, this would be the Heisenberg picture, and states would be constant in time. The interaction picture is useful for studying perturbations relative to to the behaviour of non-interacting particles, i.e. inertial matter, and implies that we are working in an inertial reference frame.
Edited on 2008-02-23 23:53:58 by CharlesFrancis
Additions:
Definition: The Hamiltonian density or interaction density is a Hermitian operator such that the interaction Hamiltonian is
Deletions:
Definition: The Hamiltonian density or interaction density is a hermitian operator such that the interaction Hamiltonian is
Edited on 2008-02-23 23:52:05 by CharlesFrancis
Additions:
Definition: The Hamiltonian density or interaction density is a hermitian operator such that the interaction Hamiltonian is
Deletions:
""<span class=math><b>Definition:</b> ; The <i>Hamiltonian density</i> or <i>interaction density</i> is a hermitian operator such that the interaction Hamiltonian is
Oldest known version of this page was edited on 2008-02-23 23:49:24 by CharlesFrancis []
Page view:
← Particle Interactions →
Interactions are modelled as a perturbation to the motion of free particles, using quantum logical OR to write the statement that, at each instant, either a particle interacts with another particle, or it does not, in which case its wave function evolves as a free particle. Relativistic considerations are used to derive the locality condition, showing that particles must meet in order to interact and which gives meaning to the claim that particles are point-like. Conservation of 3-momentum is demonstrated, showing that classical Newtonian mechanics is a consequence of the relational principles described in relativity and quantum logic.
The Interaction Hamiltonian
If interactions between particles are discrete, they will not be perfectly modelled by a continuous operator, but if the time scale for interactions is small, we might expect that time evolution will be modelled by a continuous operator on
Fock space space,
U(t) : F(t0) → F(t0 + t), to good approximation on observable timescales. In this case the arguments of section
Time Evolution apply,
Stone’s theorem can be used, and time evolution is given by
In a small time interval,
Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted
OR between uncertain possibilities, the possibility of an interaction is described by a Hamiltonian,
H, with
where
free Hamiltonian,
H0, is the Hamiltonian for non-interacting particles, and the
interaction Hamiltonian,
Hint, is a Hermitian operator describing the possibility of interaction. In general,
Hint will be a sum of terms for different types of interaction. For simplicity, I will consider only one type of interaction and spin indices will be suppressed. Spin is necessary for the full treatment of a specific interaction Hamiltonian, but has no bearing on the material on this page. Thus, the evolution of a state is given by
The Classical Correspondence
In the classical correspondence we study the behaviour of systems containing a large number,
N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are given by the expectation of the corresponding observables 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). 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, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.
Ehrenfest’s theorem describes the time evolution of a classical observable,
A .
Proof: We have, for any
,
Since
H is Hermitian,
Hence, by the definition of the
Hermitian conjugate,
where the operators act to the left. Differentiate
using the product rule,
which establishes
Ehrenfest’s theorem» and governs the classical behaviour of matter. In particular, for an observable quantity with no explicit time dependence,
Theorem: For the space indices, a = 1, 2, 3
Proof: Since space translation is the same for an observable operator,
A(x), and the corresponding classical observable,
we have
and hence,
differentiating from first principles,
The Interaction Picture
Without loss of generality, consider evolution from an initial state at time
t0 = 0. In order to study the effect of interactions it is convenient to separate interactions from free time evolution by applyinging the transformation
to states, and
to observables. This is the
interaction picture» In the absence of interaction, this would be the
Heisenberg picture, and states would be constant in time. The interaction picture is useful for studying perturbations relative to to the behaviour of non-interacting particles, i.e. inertial matter, and implies that we are working in an inertial reference frame.
In the interaction picture, the interaction Hamiltonian is
The free Hamiltonian,
H0, is unchanged in the interaction picture, since an operator commutes with a function of itself.
So, the full Hamiltonian is
H0 + HI.
Observe that
So, time evolution is given in the interaction picture by the operator,
(It is usual to overload notation by using the same symbols in different pictures. It is important to keep track of which picture is being used). Differentiate,
The Hamiltonian Density
We assume that we can define a Hermitian interaction density operator,
I(x), such that, if we were to determine the time and position at which the interaction takes place, the probability that it takes place at
x, is
. The general principle of relativity implies that
I(x) has equal effect on a matter anywhere and at any time. So, by the identification of addition with quantum logical
OR,
HI(t) can be written as a sum (as usual on this site, an integral represents a finite sum with enough terms that taking more makes no practical difference to calculation).
Definition: The Hamiltonian density or interaction density is a hermitian operator such that the interaction Hamiltonian is
<img title="The interaction Hamiltonian in terms of interaction density" alt="Interactions-14" src="images/interactions/Interactions-14g.gif" align="texttop" vspace="3">
<<
The formal construction of the Hamiltonian density, <span class=math><i>I</i>(<i>x</i>) is a major unresolved mathematical problem in quantum field theory. If particle interactions are discrete, then one might expect that a density might not exist. I will assume here that the formulae are valid in approximation for time intervals <span class="math">Δ<i>t much smaller than the resolution of measurement. I will consider the mathematical limit, <span class="math">Δ<i>t</i> → 0, and implications of its possible non-existence later.
====<a name="
ThePerturbationExpansion"></a>The Perturbation Expansion====
We have the differential equation,
<img title="derived above" alt="Interactions-15" src="images/interactions/Interactions-15.gif" align="texttop" vspace="3">
Integrate directly,
<img title="integrate" alt="Interactions-16" src="images/interactions/Interactions-16.gif" align="texttop" vspace="3">
Substituting <span class=math><i>U iteratively back into the integral gives the [[http://en.wikipedia.org/wiki/Dyson_series Dyson expansion]],
<img title="iterate" alt="Interactions-17" src="images/interactions/Interactions-17.gif" align="texttop" vspace="3">
This can also be verified by differentiating. Each term is the derivative of the next multiplied by <span class="math">−<i>iH</i>(<i>t</i>). Substituting
<img title="the interaction Hamiltonian in terms of interaction density" alt="Interactions-18" src="images/interactions/Interactions-18.gif" align="texttop" vspace="3">
gives
<img title="The perturbation expansion in terms of interaction density" alt="Interactions-19" src="images/interactions/Interactions-19.gif" align="texttop" vspace="3">
The integrals are strictly finite sums over the positions used for the basis of Hilbert space, and times <img title="integrate" alt="Interactions-20" src="images/interactions/Interactions-20.gif" align="texttop" vspace="0">.
====<a name="
TimeOrderedDiagrams"></a>Time Ordered Diagrams====
Any operator on <a href=
http://www.teleconnection.info/rqg/MultiparticleStates#IdenticalParticles>Fock» space</a>, <span class="math">F, can be written as a sum of products of <a href=
http://www.teleconnection.info/rqg/MultiparticleStates#CreationandAnnihilationOperators>creation» and annihilation operators</a>. The change of state associated with an interaction can be described as the annihilation of one state and the creation of another. Thus, a complete description of any process in interaction can be achieved through combinations of creation and annihilation operators. Expand the interaction density, <span class=math><i>I</i>(<i>x</i>), as a sum of terms of the form
<img title="a term in the interaction density" alt="Interactions-21" src="images/interactions/Interactions-21.gif" align="texttop" vspace="3">
where <img title="creation operator" alt="Interactions-22" src="images/interactions/Interactions-22.gif" align="texttop" vspace="0"> and <img title="annihilation operator" alt="Interactions-23" src="images/interactions/Interactions-23.gif" align="texttop" vspace="0"> are creation and annihilation operators for the particles and antiparticles in the interaction.
<table><td><img class="right" title="An interaction represented as a node" alt="Interactions-24" src="images/interactions/Interactions-24.gif" align="texttop" vspace="0"><span class=math><i>I</i>(<i>x</i>)</span> can be represented diagrammatically as a <a href=
http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29>vertex</a»> or node. The lines above the node correspond to creation operators, and those below the node correspond to annihilation operators.</td></table>
<img class="right" title="Time ordered diagrams for the perturbation expansion" alt="Interactions-25" src="images/interactions/Interactions-25.gif" align="texttop" vspace="0">The perturbation expansion for <img title="Inner product at time t" alt="Interactions-26" src="images/interactions/Interactions-26.gif" align="texttop" vspace="0">generates a braket between each annihilation operator, <img title="annihilation operator" alt="Interactions-27" src="images/interactions/Interactions-27.gif" align="texttop" vspace="0">, and every earlier creation operator, <img title="creation operator" alt="Interactions-28" src="images/interactions/Interactions-28.gif" align="texttop" vspace="0">, and every particle in <img title="initial state" alt="Interactions-29" src="images/interactions/Interactions-29.gif" align="texttop" vspace="0">, and a braket between every creation operator, <img title="creation operator" alt="Interactions-30" src="images/interactions/Interactions-30.gif" align="texttop" vspace="0">, and every particle in the final state, <img title="final state" alt="Interactions-31" src="images/interactions/Interactions-31.gif" align="texttop" vspace="0">, All other brakets are zero. These brakets can be represented graphically by connecting corresponding vertices. Lines representing particles are shown with arrows from bottom to top, and lines representing antiparticles with arrows from top to bottom. Then the <span class="math"><i>n</i></span><span class="normal"><sup>th</sup> term of the perturbation expansion is a sum of terms, each represented as a time-ordered [[http://en.wikipedia.org/wiki/Graph_%28mathematics%29 graph]] containing <span class="math"><i>n</i></span> vertices.
The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent <span class="math" style="font-size: 11px">OR</span> between possibities. In this interpretation, <span class=math><i>I</i>(<i>x</i>) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s [[http://en.wikipedia.org/wiki/Path_integral_formulation path integral]], or “sum over all paths” has as natural interpretation, not that a particle passes through all paths in space-time (as described by Feynman, e.g. in [[http://en.wikipedia.org/wiki/QED_%28book%29 QED: The Strange Theory of Light and Matter]], but that the sum over paths is a logical <span class="math" style="font-size: 11px">OR</span> between the possible paths that might be detected if an experiment could be done to trace the path. The time-ordered graphs give a pictorial representation of the same statement.
====<a name="
TheLocalityCondition"></a>The Locality Condition====
<<span class=math><b>Definition:</b> Let π be the permutation such that <i>t</i><sub>π(1)</sub> > <i>t</i><sub>π(2)</sub> > … <i>t</i><sub>π(<i>n</i>)</sub> . Then the <i>time-ordered product</i> is
<img title="time-ordered product" alt="Interactions-33g" src="images/interactions/Interactions-33g.gif" align="texttop" vspace="3">
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It can be seen that
<img title="time-ordered product" alt="Interactions-34" src="images/interactions/Interactions-34.gif" align="texttop" vspace="3">
Hence, we can write the perturbation expansion
<img title="rewrite the perturbation expansion using the time-ordered product" alt="Interactions-35" src="images/interactions/Interactions-35.gif" align="texttop" vspace="3">
The integrals are strictly finite sums over the positions used for the basis of Hilbert space, and times, <img title="range of time" alt="Interactions-36" src="images/interactions/Interactions-36.gif" align="texttop" vspace="0">. (Those very astute and well versed in analysis may see potential problems with this step in particular. For those more physically inclined, the justification usually given for the mathematics is that it leads to some very good physical predictions. I will not ignore that fact that it also leads to some serious inconsistencies which will have to be addressed ).
<<span class=math><b>Theorem:</b> ; (<i>Locality</i>) For any <i>x</i>, <i>y</i>, such that <i>x</i> − <i>y</i> is space-like, [<i>I</i>(<i>y</i>), <i>I</i>(<i>x</i>)] = 0.
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Under Lorentz transformation, the order of interactions, <span class=math><i>I</i>(<i>x<sub>i</sub></i>), can be changed in the time-ordered product whenever <span class=math><i>x<sub>i</sub> − y</i> is space-like. Under the condition that the initial and final states are stable states of free particles, as in scattering experiments, the calculation of probabilities cannot be affected. The locality condition follows immediately.
In the interaction picture, <a href=
http://www.teleconnection.info/rqg/Interactions#ClassicalBehaviour>Ehrenfest’s» theorem</a> states that the evolution of a classical observable quantity which is not specifically time dependent (i.e. whose value depends only on the configuration of matter) is given by the commutator of the observable with the interaction Hamiltonian (repeat the proof in the interaction picture). Observable quantities are the result of physical measurement processes, which depend on the interactions of matter, so the commutator is zero outside the light cone. It follows that no observable effects may be transmitted faster than the speed of light. Of course, this argument is circular, since it depends on the premise of special relativity that information may not travel faster than the speed of light, but it serves to illustrate the physical meaning of the locality condition.
The locality condition also gives meaning to the statement that particles are point-like objects, since it shows that showing that two particles must meet at a point in order to interact. The meaning of the integral over space in the interaction Hamiltonian is logical disjunction, <span class="math" style="font-size: 11px">OR</span>, meaning that a measurement to determine the position of the interaction could find any position, but it would always find the particles interacting at a point.
====<a name="
ConservationOfMomentum"></a>Conservation of Momentum====
<<span class=math><b>Theorem:</b> ; 3-momentum is conserved in inertial reference frames.
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This is related to [[http://en.wikipedia.org/wiki/Noether's_theorem Noether’s theorem]] and, like it, depends on invariance under space translation. A separate proof is required because this is not a [[http://en.wikipedia.org/wiki/Lagrangian Lagrangian]] formulation and an [[http://en.wikipedia.org/wiki/Action_principle action principle]] is not assumed.
For any <a href=
http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#PlaneWaveStatess>plane» wave states</a>, <img title="plane wave states after interaction" alt="Interactions-39" src="images/interactions/Interactions-39.gif" align="texttop" vspace="0">, <img title="plane wave states before interaction" alt="Interactions-40" src="images/interactions/Interactions-40.gif" align="texttop" vspace="0">
<img title="term in the perturbation expansion" alt="Interactions-41" src="images/interactions/Interactions-41.gif" align="texttop" vspace="2">
This reduces to a sum of terms containing a factor,
<img title="factor in a term in the perturbation expansion" alt="Interactions-42" src="images/interactions/Interactions-42.gif" align="texttop" vspace="2">
where <span class=math>π is a permutation of <span class=math>{1, …, <i>m</i>} and <span class=math>π<i>' is a permutation of <span class=math>{<i>m</i> + 1, …, <i>n</i>}. This is
<img title="There is factor of a delta function in each term in the perturbation expansion" alt="Interactions-43" src="images/interactions/Interactions-43.gif" align="texttop" vspace="2">
So, 3-momentum is conserved at each vertex. Between vertices, particles evolve according to the free Hamiltonian, <span class=math><i>H</i><sub>0</sub>, which also <a href=
http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>conserves» 3-momentum</a>. So, momentum is always conserved.
It is straightforward to show that if the <a href=
http://www.teleconnection.info/rqg/Evolution#Newton’sFirstLaw>mass» shell</a>"" condition is obeyed, energy is not conserved at a vertex. It will be shown in the calculation of
Feynman rules that energy is conserved for a system of interactions over large timescales.
Particle Interactions ↑ Quantum Electrodynamics →
