Most recent edit on 2008-08-13 03:47:42 by CharlesFrancis

Additions:

←  The Dirac Field Operator  ↑  →

I introduce qed by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, demonstrate that they obey locality and I define the current density observable.

Deletions:

←  The Dirac Field Operator  →

With the exception of special relativity, Quantum electrodynamics» is the most empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. I introduce the subject by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, and I define the current density observable.

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Edited on 2008-03-11 10:58:36 by CharlesFrancis

Additions:

←  The Dirac Field Operator  →

The Current Density Observable

Deletions:

←  The Dirac Field Operator  →

The Current Observable

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Edited on 2008-03-09 03:57:06 by CharlesFrancis

Additions:
With the exception of special relativity, Quantum electrodynamics» is the most empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. I introduce the subject by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, and I define the current density observable.

Deletions:
With the exception of special relativity, Quantum electrodynamics» is the most empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. I introduce the subject by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, and I define the current observable.

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Edited on 2008-03-09 03:55:47 by CharlesFrancis

Additions:
With the exception of special relativity, Quantum electrodynamics» is the most empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. I introduce the subject by constructing the Dirac field operators, which are used to describe the interactions of Dirac particles, and I define the current observable.

Deletions:
With the exception of special relativity, Quantum electrodynamics» is the most empirically accurate» theory known to established science, but it is fraught with mathematical difficulties and divergence problems. I follow a straightforward approach, and draw attention to issues which this raises. I believe that this approach gives a much truer view of underlying physics than is generally presented from quantum field theory. It is shown that the simple interaction in which a Dirac particle emits or absorbs a photon explains the form of Maxwell’s equations of electromagnetism.

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Edited on 2008-03-09 03:27:04 by CharlesFrancis

Additions:
The Dirac Field Operator ↑ The Photon Field Operator →

Deletions:
The Dirac Field Operator ↑ The Photon Field Operator →

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Edited on 2008-03-09 03:14:38 by CharlesFrancis

Additions:

←  The Dirac Field Operator  →

The Dirac Field Operator ↑ The Photon Field Operator →

Deletions:

←  Quantum Electrodynamics  →

Photons

Since the current density observable, j^a(x), is a vector, a covariant theory can be found by contracting it with another Hermitian vector operator, A^a(x). One way to do this is to introduce a particle with a spin index which transforms as a vector, and which is its own antiparticle, i.e. its creation and annihilation operators appear in the same field operator. Vector particles may have non-zero mass, but empirical evidence is that this is not so for the photon» at the limit of experimental accuracy. Zero mass is assumed.

Definition:  The photon field operator is

Definition:  The Hamiltonian density for quantum electrodynamics is

where e is an experimentally determined constant, known as charge^».
With this Hamiltonian density photons are always either created or destroyed in interaction. We cannot, therefore, talk of the measurements of the position of a photon, but only of measurement of the position at which it was annihilated, or the position at which it was created. x is not the position of a photon, but rather the position at which a charged particle would be found to have emitted or absorbed a photon, if a measurement were carried out.

RULE VIIIa.   is the formal conditional clause “If a measurement found the creation of a photon at x, …”.

RULE VIIIb.   is the formal consequent clause “…, then a measurement would find the annihilation of a photon at x”.

Definition:  The photon wave function,

is the formal statement, “if were known from previous measurement, then, in another measurement, the photon would be annihilated at x”.

Plane Wave Photon States

Since momentum is a conserved quantity, it is possible to talk about the measured momentum of a photon state. A photon created with a given momentum will be annihilated with the same momentum. So, it will be required that plane wave states are an orthogonal basis. First define a basis for spin states.

Definition:  For momentum p,

w(p, 3) is a longitudinal unit 3-vector, w(p, 3) = p ⁄ |p|,
w(p, 1) and w(p, 2) are orthogonal transverse unit vectors, so that, for r = 1, 2, 3, w(p, r) · w(p, s) = δ_rs .

The normalised spin vectors are w(p, 0) = (1, 0) and w(p, r) = (0, w(p, r)).

Definition:  For momentum p, the photon plane wave state, , in H¹ is given by the wave function,

where p² = 0 (the mass shell condition) and λ is a scalar, to be determined.
The scalar, λ, is required because the states refer to the hypothetical measurement of position of the electron which emits a photon, not the position at which a photon can be measured. Photons are always created or annihilated in interaction, and cannot be in eigenstates of a position operator. It is not meaningful to annihilate a photon at the instant of its creation. We do not require that states, , are orthogonal. Direction is determined by the distribution of matter, not by fundamental assumption, so λ depends only on the magnitude of p.
We require that is a delta function,

where η(0) = −1 and η(r) = 1 for r = 1, 2, 3. The minus sign from η(0) does not alter the expansion of the inner product for an orthonormal basis. The braket for the photon is,

The resolution of unity takes the form,

We do not have ; the braket is not positive definite, in conflict with the calculation of probabilities. In practice, we only need to generate probabilities for observations. We impose the condition that, in observations on the photon, there is no polarisation between time-like and longitudinal states,

With this restriction, probabilities for the observation time-like and longitudinal states are zero, and the braket reduces to

which is positive semidefinite». It will be seen that all four polarisation states are required for the derivation of Maxwell’s equations». We can conclude that the unobservable states have a real effect, and represent real particles, but the probability interpretation allows only the observation of a subspace containing the two transverse polarisation states, on which the inner product is positive definite. The braket is invariant under the addition of a light-like polarisation state, from which it follows that light-like polarisation cannot be determined from experimental results.
We require that the probability for the creation of a photon at x and its annihilation at y is covariant. Observe that

Then, setting

gives

which is covariant, as required.

Evolution of Photon States

We may expand using a basis of plane wave states,

Then the wave function for the state is

Since p is the momentum vector for a zero mass particle, the wave function satisfies a Klein-Gordon Equation,

Conservation of probability applies to the creation and annihilation of particles. Differentiating gives a first order equation as required by Stone’s theorem,

The Photon Field Operator

The creation operators for a plane wave state is given by . Substituting gives the photon field operator,

Theorem:  The photon field satisfies ∂²A^a = 0.
Proof: Differentiate directly.

Theorem:  For physical states, the photon field satisfies the Gupta-Bleuler gauge condition^», .
Proof: Differentiate and use absence of polarisation between light-like and longitudinal states.
Photons are Bosons, obeying commutation relations,

Substituting p → −p in the second term gives the equal time commutator,

Since the integral is invariant, the commutator is zero outside the light cone, satisfying locality.

The Locality Condition for Photons

The value of the expectation of the photon field is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A. It is required that locality is obeyed.

Theorem:  The equal time commutation relations for the photon field and its derivative obey:

Theorem (locality) :  The commutator for the photon field and its derivative is zero outside the light cone.
Proof:   Differentiating,

and

Substitute p → −p at x⁰ = y⁰. Then, for i = 1, 2, 3,

and, for the time component,

The integrals are invariant, so they are zero outside the light cone.

The Classical Field

A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density.

Definition:  The classical field, , is the expectation of the photon field operator, A(x).
It follows from Ehrenfest’s theorem that

Theorem:  The classical field, , satisfies the Lorenz gauge condition^»,

Proof:  Since the equal time commutator is zero, and using the Gupta-Bleuler gauge condition,

The Lorenz gauge condition fixes gauge up to the unobservable light-like polarisation. In classical electrodynamics one may choose a different gauge without affecting predictions, but here Lorenz gauge is fixed by the requirement of a first order covariant equation.

Classical Electromagnetism

In keeping with the idea that particles are the fundamental building blocks of matter, and have behaviour constrained by quantum theory and relativity, classical electromagnetism has not been assumed in this account. To show that classical electromagnetism is the consequence of particle interactions we need to show that is a conserved current density, and that Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.

Theorem:  The classical electromagnetic field, satisfies Maxwell’s Equations in Lorenz gauge:

Proof:
Differentiating the expectation of the photon field twice, using Ehrenfest’s theorem

Using the Hamiltonian density for qed, I(x) = ej(x) · A(x),

The equal time commutor for photons is

Maxwell’s equations in Lorenz gauge follow immediately.

Corollary:  is a classical conserved current density,

Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorenz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current density operator, and using properties of Dirac spinors.
I have given Maxwell’s equations in terms of the classical vector field, . More commonly these are expressed in terms of the components of the Faraday tensor.

Definition:  The Faraday Tensor,

has components

The standard form of Maxwell’s equations follow.

Maxwell’s Equations:

Gauss’s law:		electrostatic
Ampère-Maxwell law:		electrodynamic
Faraday’s law:		magnetodynamic
Gauss’s law:		magnetostatic

(Proof). To convert to SI units», divide space derivatives (i.e. and B) by c, put , , and use ε₀μ₀ = c.
At no point has electromagnetism been assumed. Maxwell’s equations have been found as the consequence of an underlying structure consisting of particles and simple particle interactions. The general requirements of a theory of measurement within such a structure have lead to quantum theory and relativity. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded Maxwell’s equations. To complete the demonstration that classical electromagnetism is due to the transmission of photons between charged particles we also need to show the Lorentz force law.
Quantum Electrodynamics ↑ Feynman Diagrams →

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Edited on 2008-03-09 03:03:59 by CharlesFrancis

No differences.

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Edited on 2008-03-09 01:56:22 by CharlesFrancis

No differences.

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Edited on 2008-03-09 01:54:03 by CharlesFrancis

Additions:

The Classical Field

A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density.

Definition:  The classical field, , is the expectation of the photon field operator, A(x).
It follows from Ehrenfest’s theorem that

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Edited on 2008-03-09 01:48:21 by CharlesFrancis

Additions:


Deletions:

Gauge Invariance

A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. It follows from Ehrenfest’s theorem that

This equation is unchanged under the gauge transformation , where φ^b(x) is an arbitrary solution of ∂_bφ^b = 0 and has no physical meaning. φ(x) is a gauge term and hides the true value of . Since the equations governing classical quantities dependent on A are invariant under gauge transformation, observable quantities associated with photons depend only on derivatives of .

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Edited on 2008-03-06 03:45:48 by CharlesFrancis

Additions:
The observable quantity, current density, uses the Dirac adjoint, so we expect the adjoint operators to appear alongside the field operators in the Hamiltonian density. The Dirac adjoint of the annihilation operator creates a particle,

Definition:  The current density observable is

in agreement with current density for a single particle state. For an antiparticle states, spin indices are transposed. Transposition is equivalent to pre- and post-multiplying γ by a matrix,
Since the current density observable, j^a(x), is a vector, a covariant theory can be found by contracting it with another Hermitian vector operator, A^a(x). One way to do this is to introduce a particle with a spin index which transforms as a vector, and which is its own antiparticle, i.e. its creation and annihilation operators appear in the same field operator. Vector particles may have non-zero mass, but empirical evidence is that this is not so for the photon» at the limit of experimental accuracy. Zero mass is assumed.
In keeping with the idea that particles are the fundamental building blocks of matter, and have behaviour constrained by quantum theory and relativity, classical electromagnetism has not been assumed in this account. To show that classical electromagnetism is the consequence of particle interactions we need to show that is a conserved current density, and that Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.

Corollary:  is a classical conserved current density,
Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorenz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current density operator, and using properties of Dirac spinors.

Deletions:
The observable quantity, current, uses the Dirac adjoint, so we expect the adjoint operators to appear alongside the field operators in the Hamiltonian density. The Dirac adjoint of the annihilation operator creates a particle,

Definition:  The current observable is

in agreement with current for a single particle state. For an antiparticle states, spin indices are transposed. Transposition is equivalent to pre- and post-multiplying γ by a matrix,
Since the current observable, j^a(x), is a vector, a covariant theory can be found by contracting it with another Hermitian vector operator, A^a(x). One way to do this is to introduce a particle with a spin index which transforms as a vector, and which is its own antiparticle, i.e. its creation and annihilation operators appear in the same field operator. Vector particles may have non-zero mass, but empirical evidence is that this is not so for the photon» at the limit of experimental accuracy. Zero mass is assumed.
In keeping with the idea that particles are the fundamental building blocks of matter, and have behaviour constrained by quantum theory and relativity, classical electromagnetism has not been assumed in this account. To show that classical electromagnetism is the consequence of particle interactions we need to show that is a conserved current, and that Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.

Corollary:  is a classical conserved current,
Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorenz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current operator, and using properties of Dirac spinors.

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Edited on 2008-03-03 01:12:41 by CharlesFrancis

Additions:
The value of the expectation of the photon field is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A. It is required that locality is obeyed.

Deletions:
Since the commutator of the photon field, A(x), with the interaction Hamiltonian is zero, its expectation is constant by Ehrenfest’s theorem. Its value is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A. It is required that locality is obeyed.

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Edited on 2008-02-29 05:25:30 by CharlesFrancis

Additions:

Theorem:  The classical field, , satisfies the Lorenz gauge condition^»,

The Lorenz gauge condition fixes gauge up to the unobservable light-like polarisation. In classical electrodynamics one may choose a different gauge without affecting predictions, but here Lorenz gauge is fixed by the requirement of a first order covariant equation.

Theorem:  The classical electromagnetic field, satisfies Maxwell’s Equations in Lorenz gauge:

Maxwell’s equations in Lorenz gauge follow immediately.
Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorenz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current operator, and using properties of Dirac spinors.

Deletions:

Theorem:  The classical field, , satisfies the Lorentz gauge condition^»,

The Lorentz gauge condition fixes gauge up to the unobservable light-like polarisation. In classical electrodynamics one may choose a different gauge without affecting predictions, but here Lorentz gauge is fixed by the requirement of a first order covariant equation.

Theorem:  The classical electromagnetic field, satisfies Maxwell’s Equations in Lorentz gauge:

Maxwell’s equations in Lorentz gauge follow immediately.
Proof: Partial derivatives commute (Clairaut’s Theorem). Differentiate Maxwell’s equations and use the Lorentz gauge condition. This can also be proved directly by calculating the commutator of the Hamiltonian with the current operator, and using properties of Dirac spinors.

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Edited on 2008-02-29 01:37:09 by CharlesFrancis

Additions:

We may expand using a basis of plane wave states,

Then the wave function for the state is

The creation operators for a plane wave state is given by . Substituting gives the photon field operator,

Theorem:  For physical states, the photon field satisfies the Gupta-Bleuler gauge condition^», .

This equation is unchanged under the gauge transformation , where φ^b(x) is an arbitrary solution of ∂_bφ^b = 0 and has no physical meaning. φ(x) is a gauge term and hides the true value of . Since the equations governing classical quantities dependent on A are invariant under gauge transformation, observable quantities associated with photons depend only on derivatives of .

Deletions:

We may expand using a basis of plane wave states,

Then the wave function for the state is

The creation operators for a plane wave state is given by . Substituting gives the photon field operator,

Theorem:  For physical states, the photon field satisfies the Gupta-Bleuler gauge condition^», .

This equation is unchanged under the gauge transformation , where φ^b(x) is an arbitrary solution of ∂_bφ^b = 0 and has no physical meaning. φ(x) is a gauge term and hides the true value of . Since the equations governing classical quantities dependent on A are invariant under gauge transformation, observable quantities associated with photons depend only on derivatives of .

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Edited on 2008-02-27 04:57:40 by CharlesFrancis

Additions:
Using the Hamiltonian density for qed, I(x) = ej(x) · A(x),

Deletions:
Using the Hamiltonian density for qed, I = ej(x) · A(x),

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Edited on 2008-02-27 04:56:06 by CharlesFrancis

Additions:
Using the Hamiltonian density for qed, I = ej(x) · A(x),

Deletions:
Using the Hamiltonian density for qed, I = ej^a(x)A(x),

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Edited on 2008-02-27 04:53:38 by CharlesFrancis

Additions:
Using the Hamiltonian density for qed, I = ej^a(x)A(x),

Deletions:
Using the Hamiltonian density for qed, ""<span class=math><i>I</i> = <i>ej<sup>a</a></a>(<i>x</i>)<i>A</i>(<i>x</i>)",

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Edited on 2008-02-27 04:26:39 by CharlesFrancis

Additions:

Theorem (locality):  The anticommutation relation for the Dirac field and the Dirac adjoint is zero outside the light cone.
where e is an experimentally determined constant, known as charge^».
Since the commutator of the photon field, A(x), with the interaction Hamiltonian is zero, its expectation is constant by Ehrenfest’s theorem. Its value is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A. It is required that locality is obeyed.

Theorem:  The equal time commutation relations for the photon field and its derivative obey:

Theorem (locality) :  The commutator for the photon field and its derivative is zero outside the light cone.
Using the Hamiltonian density for qed, ""<span class=math><i>I</i> = <i>ej<sup>a</a></a>(<i>x</i>)<i>A</i>(<i>x</i>)",

Deletions:

Theorem:  The anticommutation relation for the Dirac field and the Dirac adjoint is zero outside the light cone.
where e is an experimentally determined constant, known as charge.
Since the commutator of the photon field, A(x), with the interaction Hamiltonian is zero, its expectation is constant by Ehrenfest’s theorem. Its value is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A.

Theorem:  The commutator satisfies the locality condition that is zero outside the light cone and
Using the Hamiltonian density for qed

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Edited on 2008-02-24 04:24:58 by CharlesFrancis

Additions:
Using the Hamiltonian density for qed

Deletions:
Using the Hamiltonian density for qed
""<a href=http://www.teleconnection.info/rqg/QED#Photons»

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Oldest known version of this page was edited on 2008-02-24 03:17:38 by CharlesFrancis []

←  Quantum Electrodynamics  →

Quantum Field Operators

We may expect symmetry between particles and antiparticles. The interpretation of antiparticles as negative energy particles going backwards in time motivates the definition of the quantum field operator».


We will find stronger reasons, to do with satisfying locality, for thinking that interaction operators are composed of field operators with this general form. The Hermitian conjugate of a quantum field operator, has the reverse effect, creating a particle or annihilating an antiparticle,

While quantum field operators defined here are formally those of standard quantum field theory», the interpretation adheres more closely to the views of Dirac and Feynman than to the “modern view” that fields are the fundamental objects of Nature. I have not used canonical quantization» to convert from a classical theory or second quantization^» of a field.

Normal Order

The interaction of a particle will be modelled by the annihilation of the old state of the particle and the creation of a new state. This means that all annihilation operators must be placed to the right of all creation operators in the interaction density, I(x). An operator for which this is so is in normal order». Putting an operator into normal order is normal ordering, and is denoted by colons.


In particular, the normal ordered product of fields is
where a plus is used for Bosons and a minus for Fermions.

The Dirac Field Operator

Since we can measure the position of an electron, it must be possible to form a projection operator for position at given time,
from the Hamiltonian density and a suitable configuration of matter (more accurately, X_P(x) is summed over a small range of positions depending on the resolution of measurement). We may expect symmetry between particles and antiparticles. The interpretation of antiparticles as negative energy particles going backwards in time indicates the replacement of negative energy annihilation operators with creation operators. This motivates the definition of the Dirac field operator», which annihilates a particle or creates the corresponding antiparticle.


The observable quantity, current, uses the Dirac adjoint, so we expect the adjoint operators to appear alongside the field operators in the Hamiltonian density. The Dirac adjoint of the annihilation operator creates a particle,
The Dirac adjoint of the antiparticle creation operator annihilates an antiparticle,
The Dirac adjoint of the field operator creates a particle or annihilates an antiparticle,

Locality of Dirac Field Operators

Since Dirac particles are Fermions we have anticommutation relations for the Dirac field operator.
Dirac field operators will appear in pairs in the Hamiltonian density in such a way as to ensure commutation relations required of the locality condition.


Proof.

The Current Observable

For electrons, current is an observable quantity. Since measurement is always the result of interactions between matter, a Hermitian operator, j, whose expectation is the classical electrical current must appear in the Hamiltonian density. To ensure that locality is satisfied, current is composed of Dirac field operators.


Then, given the particle state in H¹ with ,
in agreement with current for a single particle state. For an antiparticle states, spin indices are transposed. Transposition is equivalent to pre- and post-multiplying γ by a matrix,
which has the effect on the wave function of reversing the order of the spin indices. Thus, a negative energy spin down electron will appear as a positive energy spin up positron.

Photons

Since the current observable, j^a(x), is a vector, a covariant theory can be found by contracting it with another Hermitian vector operator, A^a(x). One way to do this is to introduce a particle with a spin index which transforms as a vector, and which is its own antiparticle, i.e. its creation and annihilation operators appear in the same field operator. Vector particles may have non-zero mass, but empirical evidence is that this is not so for the photon» at the limit of experimental accuracy. Zero mass is assumed.


With this Hamiltonian density photons are always either created or destroyed in interaction. We cannot, therefore, talk of the measurements of the position of a photon, but only of measurement of the position at which it was annihilated, or the position at which it was created. x is not the position of a photon, but rather the position at which a charged particle would be found to have emitted or absorbed a photon, if a measurement were carried out.

Plane Wave Photon States

Since momentum is a conserved quantity, it is possible to talk about the measured momentum of a photon state. A photon created with a given momentum will be annihilated with the same momentum. So, it will be required that plane wave states are an orthogonal basis. First define a basis for spin states.


The scalar, λ, is required because the states refer to the hypothetical measurement of position of the electron which emits a photon, not the position at which a photon can be measured. Photons are always created or annihilated in interaction, and cannot be in eigenstates of a position operator. It is not meaningful to annihilate a photon at the instant of its creation. We do not require that states, , are orthogonal. Direction is determined by the distribution of matter, not by fundamental assumption, so λ depends only on the magnitude of p.

We require that is a delta function,
where η(0) = −1 and η(r) = 1 for r = 1, 2, 3. The minus sign from η(0) does not alter the expansion of the inner product for an orthonormal basis. The braket for the photon is,
The resolution of unity takes the form,
We do not have ; the braket is not positive definite, in conflict with the calculation of probabilities. In practice, we only need to generate probabilities for observations. We impose the condition that, in observations on the photon, there is no polarisation between time-like and longitudinal states,
With this restriction, probabilities for the observation time-like and longitudinal states are zero, and the braket reduces to
which is positive semidefinite». It will be seen that all four polarisation states are required for the derivation of Maxwell’s equations». We can conclude that the unobservable states have a real effect, and represent real particles, but the probability interpretation allows only the observation of a subspace containing the two transverse polarisation states, on which the inner product is positive definite. The braket is invariant under the addition of a light-like polarisation state, from which it follows that light-like polarisation cannot be determined from experimental results.

We require that the probability for the creation of a photon at x and its annihilation at y is covariant. Observe that
Then, setting
gives
which is covariant, as required.

Evolution of Photon States

We may expand using a basis of plane wave states,
Then the wave function for the state is
Since p is the momentum vector for a zero mass particle, the wave function satisfies a Klein-Gordon Equation,
Conservation of probability applies to the creation and annihilation of particles. Differentiating gives a first order equation as required by Stone’s theorem,

The Photon Field Operator

The creation operators for a plane wave state is given by . Substituting gives the photon field operator,


Proof: Differentiate directly.


Proof: Differentiate and use absence of polarisation between light-like and longitudinal states.

Photons are Bosons, obeying commutation relations,
Substituting p → −p in the second term gives the equal time commutator,
Since the integral is invariant, the commutator is zero outside the light cone, satisfying locality.

Gauge Invariance

A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. It follows from Ehrenfest’s theorem that
This equation is unchanged under the gauge transformation , where φ^b(x) is an arbitrary solution of ∂_bφ^b = 0 and has no physical meaning. φ(x) is a gauge term and hides the true value of . Since the equations governing classical quantities dependent on A are invariant under gauge transformation, observable quantities associated with photons depend only on derivatives of .


Proof:  Since the equal time commutator is zero, and using the Gupta-Bleuler gauge condition,
The Lorentz gauge condition fixes gauge up to the unobservable light-like polarisation. In classical electrodynamics one may choose a different gauge without affecting predictions, but here Lorentz gauge is fixed by the requirement of a first order covariant equation.

The Locality Condition for Photons

Since the commutator of the photon field, A(x), with the interaction Hamiltonian is zero, its expectation is constant by Ehrenfest’s theorem. Its value is hidden by gauge invariance. As a result, physical laws depend on derivatives of the photon field, not directly on A.

Proof:   Differentiating,
and
Substitute p → −p at x⁰ = y⁰. Then, for i = 1, 2, 3,
and, for the time component,
The integrals are invariant, so they are zero outside the light cone.

Classical Electromagnetism

In keeping with the idea that particles are the fundamental building blocks of matter, and have behaviour constrained by quantum theory and relativity, classical electromagnetism has not been assumed in this account. To show that classical electromagnetism is the consequence of particle interactions we need to show that is a conserved current, and that Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.


Proof:
Differentiating the expectation of the photon field twice, using Ehrenfest’s theorem
Using the Hamiltonian density for qed
<img title="Ehrenfest’s theorem" alt="QED-113" src="images/qed/QED-113.gif" align="texttop" vspace="3"> The <a href=http://www.teleconnection.info/rqg/QED#TheLocalityConditionForPhotons>equal» time commutor</a> for photons is <img title="equal time commutator" alt="QED-114" src="images/qed/QED-114.gif" align="texttop" vspace="3"> Maxwell’s equations in Lorentz gauge follow immediately. <<span class=math><b>Corollary:</b> ; <img title="classical current" alt="QED-101" src="images/qed/QED-101g.gif" align="texttop" vspace="0"> is a classical conserved current, <img title="conservation equation" alt="QED-102" src="images/qed/QED-102g.gif" align="texttop" vspace="3"> << **Proof:** Partial derivatives commute (<a href=http://www.teleconnection.info/rqg/Operators#Clairaut’sTheorem>Clairaut’s» Theorem</a>). Differentiate Maxwell’s equations and use the Lorentz gauge condition. This can also be <a href=http://www.teleconnection.info/rqg/QEDSub#ConservedCurrent>proved» directly</a> by calculating the commutator of the Hamiltonian with the current operator, and using properties of Dirac spinors. I have given Maxwell’s equations in terms of the classical vector field, <img title="equal time commutator" alt="QED-115" src="images/qed/QED-115.gif" align="texttop" vspace="0">. More commonly these are expressed in terms of the components of the <a href=http://www.teleconnection.info/rqg/IntroductionToTensors#ElectromagneticForce>Faraday» tensor</a>. <<span class=math><b>Definition:</b> ; The <i>Faraday Tensor</i>, <img title="equal time commutator" alt="QED-116" src="images/qed/QED-116g.gif" align="texttop" vspace="3"> has components <img title="The Faraday Tensor" alt="QED-117" src="images/qed/QED-117g.gif" align="texttop" vspace="2"> << <a name=Maxwell’sEquations></a>The standard form of Maxwell’s equations follow. <<span class=math><b>Maxwell’s Equations: <span class=math><table><td valign=top>Gauss’s law:</td><td>  </td><td valign=top>electrostatic</td><td>   </td><td><img title="The electrostatic law" alt="QED-118" src="images/qed/QED-118g.gif" align="texttop" vspace="3"> </td><tr>
<td valign=top>Ampère-Maxwell law:</td><td>  </td><td valign=top>electrodynamic</td><td></td><td><img title="The electrodynamic law" alt="QED-119" src="images/qed/QED-119g.gif" align="texttop" vspace="4"></td><tr>
<td valign=top>Faraday’s law:</td><td>  </td><td valign=top>magnetodynamic</td><td></td><td><img title="The magnetodynamic law" alt="QED-120" src="images/qed/QED-120g.gif" align="texttop" vspace="3"></td><tr>
<td valign=top>Gauss’s law:</td><td>  </td><td valign=top>magnetostatic</td><td></td><td><img title="magnetostatic law" alt="QED-121" src="images/qed/QED-121g.gif" align="texttop" vspace="3"></td></table> << (<a href=http://www.teleconnection.info/rqg/QEDSub#Maxwell’sEquations>Proof</a>"")». To convert to SI units», divide space derivatives (i.e. and B) by c, put , , and use ε₀μ₀ = c.

At no point has electromagnetism been assumed. Maxwell’s equations have been found as the consequence of an underlying structure consisting of particles and simple particle interactions. The general requirements of a theory of measurement within such a structure have lead to quantum theory and relativity. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded Maxwell’s equations. To complete the demonstration that classical electromagnetism is due to the transmission of photons between charged particles we also need to show the Lorentz force law.

Quantum Electrodynamics ↑ Feynman Diagrams →