Most recent edit on 2008-08-13 03:50:23 by CharlesFrancis

Additions:

← Classical Electromagnetism ↑ →

Since laws of physics must obey general covariance, the parameter τ is the proper time of the matter on which the force acts, not coordinate time defined by a particular observer. The electromagnetic force on a charged particle will be evaluated in the rest frame of the particle, in which CEM-28

and the current is CEM-29

. In the rest frame,

Deletions:

← Classical Electromagnetism →

and the current is CEM-29

. In the rest frame,

Edited on 2008-03-29 00:44:28 by CharlesFrancis

Additions:
After Lorentz transformation, this is

Deletions:
After Lorentz transformation this is

Edited on 2008-03-29 00:38:36 by CharlesFrancis

Additions:
With this definition, 3-vector force obeys the Lorentz force law.

Deletions:
With this definition, 3-vector Force obeys the Lorentz force law.

Edited on 2008-03-29 00:27:21 by CharlesFrancis

Additions:
With this definition, 3-vector Force obeys the Lorentz force law.

Deletions:
With these definitions, we have that the 3-vector Force obeys the Lorentz force law.

Edited on 2008-03-29 00:25:58 by CharlesFrancis

Additions:
where F^ab is the Faraday tensor. The classical E and B fields are defined as components of the Faraday tensor.

Deletions:
where F^ab is the Faraday tensor.

The E and B fields are defined as components of the Faraday tensor.

Definition: The components of the Faraday Tensor are,

Edited on 2008-03-29 00:18:16 by CharlesFrancis

Additions:

Definition: The components of the Faraday Tensor are,

Deletions:

Definition: The Faraday Tensor,

Edited on 2008-03-29 00:14:32 by CharlesFrancis

Additions:
where F^ab is the Faraday tensor.

The E and B fields are defined as components of the Faraday tensor.

Definition: The Faraday Tensor,
With these definitions, we have that the 3-vector Force obeys the Lorentz force law.

Deletions:
This motivates the definition of the Faraday tensor.

Definition: The classical E and B fields are defined as components of the Faraday Tensor,
With these definitions we have shown the Lorentz force law.

Edited on 2008-03-28 10:50:55 by CharlesFrancis

Additions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that CEM-1

is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.

Definition: The classical field, CEM-2

, is the expectation of the photon field operator, A(x).

Theorem: The classical field, CEM-3

, satisfies the Lorenz gauge condition^»,

Theorem: The classical electromagnetic field, CEM-61

satisfies Maxwell’s equations in Lorenz gauge:

Corollary: CEM-66

is a classical conserved current density,

I have given Maxwell’s equations in terms of the classical vector field, CEM-68

. The standard form of Maxwell’s equations follow.

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. CEM-122

and B) by c, put CEM-74

, and use ε₀μ₀ = c.

Deletions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that QED-94

is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.

Definition: The classical field, QED-81

, is the expectation of the photon field operator, A(x).

Theorem: The classical field, QED-81

, satisfies the Lorenz gauge condition^»,

Theorem: The classical electromagnetic field, QED-111

satisfies Maxwell’s equations in Lorenz gauge:

Corollary: QED-101

is a classical conserved current density,

I have given Maxwell’s equations in terms of the classical vector field, QED-115

. The standard form of Maxwell’s equations follow.

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. QED-122

and B) by c, put QED-123

, and use ε₀μ₀ = c.

Edited on 2008-03-13 10:51:59 by CharlesFrancis

Additions:
A(x) has the required property of an observable, that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law» and Maxwell’s equations».
The term gauge symmetry» is something of a misnomer. It was introduced by Herman Weyl», as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, Vladimir Fock» and Fritz London» adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown Gauge Invariance.

Deletions:
A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law» and Maxwell’s equations».
The term gauge symmetry» is something of a misnomer. It was introduced by Herman Weyl», as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, Vladimir Fock» and Fritz London» adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown GaugeInvariance.

Edited on 2008-03-11 08:27:38 by CharlesFrancis

Additions:
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation (courtesy of Hsin-Chia Cheng»; for a general discussion, see Costella & McKellar^»). For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.

Deletions:
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^» (courtesy of Hsin-Chia Cheng»; for a general discussion, see Costella & McKellar). For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.

Edited on 2008-03-11 03:22:13 by CharlesFrancis

Additions:

The Interacting Dirac Equation:

Theorem: The Lorentz Force law,

Deletions:

The Interacting Dirac Equation:

""Theorem: ; The Lorentz Force law,

Edited on 2008-03-11 03:20:01 by CharlesFrancis

Additions:
A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law» and Maxwell’s equations».
The local phase transformation,
applied to the field operators, makes no difference to the current and so leaves the predictions of the theory unchanged (equivalently the transformation may be applied to the creation operators, remembering the sign change for antiparticles).

Definition: The freedom to vary the phase of the field operators is the gauge symmetry of qed^».
The term gauge symmetry» is something of a misnomer. It was introduced by Herman Weyl», as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, Vladimir Fock» and Fritz London» adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown GaugeInvariance.

Momentum in the Interacting Theory

This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^» (courtesy of Hsin-Chia Cheng»; for a general discussion, see Costella & McKellar). For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
Since laws of physics must obey general covariance, the parameter τ is the proper time of the matter on which the force acts, not coordinate time defined by a particular observer. The electromagnetic force on a charged particle will be evaluated in the rest frame of the particle, in which CEM-28

and the current is CEM-29

. In the rest frame,

Theorem: The Lorentz Force law, can be written, in terms of creation operators acting on any ket <img title="a ket" alt="CEM-37" src="images/cem/CEM-37.gif" align="texttop" vspace="0">, A <a href=http://www.teleconnection.info/rqg/CEM#GaugeSymmetry>local» gauge transformation</a>"" applied to the creation operators,
At no point has electromagnetism been assumed, but rather it has been found from an underlying structure consisting only of particles and interactions. The general requirements of a theory of measurement within such a structure have lead to relativity and to quantum theory. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded a conserved current, the Lorentz force law, and Maxwell’s equations. The new predictions of quantum electrodynamics come from the calculation of Feynman diagrams; it is sometimes suggested that qed is only valid as perturbation theory, but the derivations of the classical equations shows that it is, as Feynman described, a complete description of the interactions underlying electromagnetism.

Deletions:
A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law and Maxwell’s equations.
If one applies a local phase transformation
to the field operators, this will make no difference to the current and the predictions of the theory will be unchanged (equivalently one may apply the transformation to the creation operators, remembering the sign change for antiparticles).

Momentum in the interacting Theory

This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^» (courtesy of Hsin-Chia Cheng». For a general discussion, see Costella & McKellar. For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
Since laws of physics must obey general covariance, the parameter τ is the proper time of the matter on which the force acts, not coordinate time defined by a particular observer. The electromagnetic force on a charged particle will be evaluated in the rest frame of the particle, in which CEM-28

and the current is CEM-29

. In the rest frame,

Theorem: The Lorentz Force law,
can be written, in terms of creation operators, acting on any ket CEM-37

A local gauge transformation applied to the creation operators,
At no point has electromagnetism been assumed, but rather it has been found from an underlying structure consisting only of particles and interactions. The general requirements of a theory of measurement within such a structure have lead to relativity and to quantum theory. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded a conserved current, the Lorentz force law, and Maxwell’s equations.

Edited on 2008-03-11 00:08:36 by CharlesFrancis

Additions:
Proof: By locality the equal time commutator is zero. Using the Gupta-Bleuler gauge condition,
Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the interaction density

Gauge Symmetry

Momentum in the interacting Theory

but states evolve according to the full Hamiltonian, whereas the creation operators are defined on the Fock space of non-interacting particles, and create states obeying the Dirac equation. There is a real phase shift corresponding to change in momentum, which must be distinguished from the arbitrary phase in the definition of field operators.
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^» (courtesy of Hsin-Chia Cheng». For a general discussion, see Costella & McKellar. For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
Thus the expectation, CEM-18

, of the operator which creates and annihilates photons, acts in the manner of a classical vector field, modifying energy and momentum. This is the standard formula for momentum in the presence of a field, but normally it would be assumed, on phenomenological grounds, i.e. because we know empirically that it gives correct predictions. A phenomenological treatment assumes the concept of a classical field defined on a spacetime background. The treatment here has assumed only particles, electrons and photons, and has arrived at the classical field in an approximation in which it is an expectation of photon creation/annihilation.
With the replacement of the momentum operator for non-interacting particles with the corresponding operator taking interactions into account,
the Dirac equation,

Working in the field picture, we have, from Ehrenfest’s theorem,
where the product rule of differentiation has been used to find the second term.

A local gauge transformation applied to the creation operators,
Differentiating the expectation of the photon field twice, using Ehrenfest’s theorem,
At no point has electromagnetism been assumed, but rather it has been found from an underlying structure consisting only of particles and interactions. The general requirements of a theory of measurement within such a structure have lead to relativity and to quantum theory. In turn, this has lead to spin-½ Dirac particles and the vector photon. The most straightforward interaction between these particles has yielded a conserved current, the Lorentz force law, and Maxwell’s equations.

Deletions:
Proof: Since the equal time commutator is zero, and using the Gupta-Bleuler gauge condition,
Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the interaction density
but states, CEM-11

evolve according to the full Hamiltonian, whereas the creation operators are defined on the Fock space of non-interacting particles, and create states obeying the Dirac equation. There is a real phase shift corresponding to change in momentum, which must be distinguished from the arbitrary phase in the definition of field operators.
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^». For a general discussion, see Costella & McKellar Am.J.Phys. 63 (1995) 1119 ^». For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
Thus the expectation, CEM-18

, of the operator which creates and annihilates photons, acts in the manner of a classical vector field, modifying energy and momentum. This is the standard formula for momentum in the presence of a field, but normally it would be assumed, on phenomenological grounds, i.e. because we know it works. For that one must also assume a concept of a classical field defined on a spacetime background. The treatment here has assumed only particles, electrons and photons, and has arrived at the classical field in an approximation in which it is an expectation of photon creation/annihilation.
With the replacement
The Dirac equation

Working in the field picture, we have, from Ehrenfest’s theorem
where the product rule of differentiation is used to find the second term.

A local gauge transformation applied to the creation operators
Differentiating the expectation of the photon field twice, using Ehrenfest’s theorem
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.

Edited on 2008-03-10 11:57:08 by CharlesFrancis

Additions:
Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the interaction density
where j is the current observable,

Definition: The freedom to vary the phase of the field operators is the gauge symmetry of qed^».
extracts the frequency and wavelength of the the wave function. When the simplified notation, i∂^a, is used as an operator on ket space, the integral, the bra, and the ket are implicit. We would like to use Ehrenfest’s theorem to calculate the classical force due to the interaction, by differentiating the expectation of momentum,
In the semi-classical correspondence, for small t, this may be treated as a perturbation to the evolution of a non-interacting particle, in which the interaction is replaced with an expectation. For a classical particle with position x and velocity CEM-14

, the classical current is CEM-15

. The expectation of the interaction Hamiltonian is
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation^». For a general discussion, see Costella & McKellar Am.J.Phys. 63 (1995) 1119 ^». For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
becomes the interacting Dirac equation^».
can be written, in terms of creation operators, acting on any ket CEM-37

Deletions:
Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the <a href=http://www.teleconnection.info/rqg/QED#Photons>interaction» density</a>
where j is the <a href=http://www.teleconnection.info/rqg/QED#TheCurrentObservable>current» observable,

Definition: The freedom to vary the phase of the field operators is the gauge symmetry of qed.
extracts the frequency and wavelength of the the wave function. When the simplified notation, i∂^a, is used as an operator on ket space, the integral, the bra and the ket are implicit. We would like to use Ehrenfest’s theorem to calculate the classical force due to the interaction, by differentiating the expectation of momentum,
In the semi-classical correspondence, for small t, this may be treated as a perturbation to the evolution of a non-interacting particle, in which the interaction is replaced with an expectation. For a classical particle with position x and velocity CEM-14

, the classical current is CEM-15

. The expectation of the interaction Hamiltonian is
This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation. For a general discussion, see Costella & McKellar Am.J.Phys. 63 (1995) 1119 . For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
becomes the <a href=http://en.wikipedia.org/wiki/Dirac_equation#Coupling_to_an_electromagnetic_field>interacting» Dirac equation</a>
can be written, in terms of creation operators, acting on any ket CEM-37

Edited on 2008-03-10 11:41:15 by CharlesFrancis

Additions:
Observable quantities are determined through measurement, i.e. through interaction with other matter. Interactions are described by the <a href=http://www.teleconnection.info/rqg/QED#Photons>interaction» density</a>
where j is the <a href=http://www.teleconnection.info/rqg/QED#TheCurrentObservable>current» observable,

If one applies a local phase transformation

to the field operators, this will make no difference to the current and the predictions of the theory will be unchanged (equivalently one may apply the transformation to the creation operators, remembering the sign change for antiparticles).

Definition: The freedom to vary the phase of the field operators is the gauge symmetry of qed.
The term gauge symmetry» is something of a misnomer. It was introduced by Herman Weyl», as part of an attempt to extend the local scale invariance of general relativity to unification with electrodynamics. That attempt failed, but later Weyl, Vladimir Fock» and Fritz London» adapted the idea and applied it to phase symmetry in quantum theory, and it is to phase symmetry that the term now applies. The relation to the of this phase symmetry to a corresponding symmetry in classical electrodynamics is shown below.
In the absence of interactions, there is no issue with local gauge freedom. The phase of an electron wave function is fixed at the point of creation and becomes simply the global symmetry of the one particle theory, in which kets can be multiplied by constant phase without altering their meaning in quantum logic. When interactions are introduced the result is that the evolution of the wave function does not match the evolution of the field operator which created it, and which is defined on the non-interacting space. A difficulty arises because the momentum observable in the non-interacting theory,

extracts the frequency and wavelength of the the wave function. When the simplified notation, i∂^a, is used as an operator on ket space, the integral, the bra and the ket are implicit. We would like to use Ehrenfest’s theorem to calculate the classical force due to the interaction, by differentiating the expectation of momentum,

but states, CEM-11

In the field picture states evolve as in the Schrödinger picture for non-interacting particles. The momentum operator in the field picture is

In the semi-classical correspondence, for small t, this may be treated as a perturbation to the evolution of a non-interacting particle, in which the interaction is replaced with an expectation. For a classical particle with position x and velocity CEM-14

, the classical current is CEM-15

. The expectation of the interaction Hamiltonian is

This may be done more precisely, taking spin and antiparticle states into account, using the Foldy-Wouthuysen Transformation. For a general discussion, see Costella & McKellar Am.J.Phys. 63 (1995) 1119 . For the present treatment, I will merely show the Lorentz force law for particles, ignoring spin.
Replacing the interaction Hamiltonian with its expectation, the momentum operator in the field picture is

Thus the expectation, CEM-18

, of the operator which creates and annihilates photons, acts in the manner of a classical vector field, modifying energy and momentum. This is the standard formula for momentum in the presence of a field, but normally it would be assumed, on phenomenological grounds, i.e. because we know it works. For that one must also assume a concept of a classical field defined on a spacetime background. The treatment here has assumed only particles, electrons and photons, and has arrived at the classical field in an approximation in which it is an expectation of photon creation/annihilation.

The Interacting Dirac Equation

With the replacement

The Dirac equation

becomes the <a href=http://en.wikipedia.org/wiki/Dirac_equation#Coupling_to_an_electromagnetic_field>interacting» Dirac equation</a>

The Interacting Dirac Equation:

The interacting Dirac equation describes the behaviour of an electron in a classical field, and predicts a value of 2 for the gyromagnetic moment^», or g-factor», of the electron. Higher order corrections leading to a slightly larger value may be calculated from Feynman diagrams, and agree with the measured value to 14 significant figures.

The Lorentz Force Law

Working in the field picture, we have, from Ehrenfest’s theorem

Using expectations for the interaction, as above, we have

Substituting for H, together with

and dropping the suffix F (the expectation is the same in any picture),

where the product rule of differentiation is used to find the second term.
Classical force is defined as the rate of change of momentum, according to Newton’s second law.

and the current is CEM-29

. In the rest frame,

After Lorentz transformation this is

This motivates the definition of the Faraday tensor.

Definition: The classical E and B fields are defined as components of the Faraday Tensor,

With these definitions we have shown the Lorentz force law.

Theorem: The Lorentz Force law,

Gauge Invariance

The interacting Dirac equation,

can be written, in terms of creation operators, acting on any ket CEM-37

A local gauge transformation applied to the creation operators

gives

So,

which is identical to the original form of the interacting Dirac equation apart from the replacement

But the Faraday tensor is also unchanged by this replacement,

So the local phase symmetry of the field operators is precisely equivalent to the well known symmetry of the classical electromagnetic field.

Maxwell’s Equations

Theorem: The classical electromagnetic field, QED-111

satisfies Maxwell’s equations in Lorenz gauge:

Using the Hamiltonian density for qed, I(x) = ej(x) · A(x),
The equal time commutor for photons is
I have given Maxwell’s equations in terms of the classical vector field, QED-115

. The standard form of Maxwell’s equations follow.

Deletions:

Classical Electromagnetism

Theorem: The classical electromagnetic field, QED-111

satisfies Maxwell’s Equations in Lorenz gauge:

Using the Hamiltonian density for qed, I(x) = ej(x) · A(x),
The equal time commutor for photons is
I have given Maxwell’s equations in terms of the classical vector field, QED-115

. More commonly these are expressed in terms of the components of the Faraday tensor.

has components

The standard form of Maxwell’s equations follow.

Edited on 2008-03-09 04:30:47 by CharlesFrancis

Additions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that QED-94

is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.
A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density. A classical field can be defined as the expectation of A. To establish that this is the classical electromagnetic field, it will be necessary to establish the Lorentz force law and Maxwell’s equations.

Deletions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that j is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.
A(x) has the required properties of an observable that it is a Hermitian operator appearing in the Hamiltonian density.
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 QED-94

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.

Edited on 2008-03-09 04:16:52 by CharlesFrancis

Additions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that j is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the minimal interaction in which a photon is emitted from, or absorbed by, a Dirac particle.

Deletions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that j is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.

Edited on 2008-03-09 04:16:01 by CharlesFrancis

Additions:
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 see that classical electromagnetism is the consequence of particle interactions we need to show that j is a conserved current, and that the Lorentz force law and Maxwell’s equations follow from the interaction in which a photon is emitted from or absorbed by the a Dirac particle.

Oldest known version of this page was edited on 2008-03-09 03:34:30 by CharlesFrancis []

Page view:

← Classical Electromagnetism →

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, QED-81

, is the expectation of the photon field operator, A(x).

It follows from Ehrenfest’s theorem that

Theorem: The classical field, QED-81

, 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 QED-94

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, QED-111

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: QED-101

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, QED-115

. 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. QED-122

and B) by c, put QED-123

, 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.

Classical Electromagnetism ↑ Feynman Diagrams →

RQG : CEM

← Classical Electromagnetism ↑ →

← Classical Electromagnetism →

Momentum in the Interacting Theory

Momentum in the interacting Theory

Gauge Symmetry

Momentum in the interacting Theory

The Interacting Dirac Equation

The Lorentz Force Law

Gauge Invariance

Maxwell’s Equations

Classical Electromagnetism

← Classical Electromagnetism →

The Classical Field

Classical Electromagnetism