Most recent edit on 2008-03-28 10:48:25 by CharlesFrancis

Additions:

Maxwell’s equations
Lorenz gauge
Faraday tensor

Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws.

Deletions:

Maxwell’s equations
Lorenz gauge
Faraday tensor

Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws.

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Edited on 2008-03-10 06:50:35 by CharlesFrancis

Additions:
Return

Deletions:
Return

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Edited on 2008-03-09 03:36:25 by CharlesFrancis

Additions:
Return
Return

Deletions:
Return
Return

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Edited on 2008-03-06 03:49:02 by CharlesFrancis

Additions:

Theorem:  is a classical conserved current density,
Proof of theorem:  We have, from Ehrenfest’s theorem, that the expectation of the current density obeys
Current density is given by

Deletions:

Theorem:  is a classical conserved current,

Proof of theorem:  We have, from Ehrenfest’s theorem, that the expectation of the current obeys
Current is given by

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Edited on 2008-03-03 01:16:07 by CharlesFrancis

Additions:
Proof of theorem:  We have, from Ehrenfest’s theorem, that the expectation of the current obeys

Deletions:
Proof of theorem:  We have, from Ehrenfest’s theorem, that a classical quantity obeys

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Edited on 2008-02-29 05:27:37 by CharlesFrancis

Additions:
<td valign=top>Lorenz gauge</td><td>  </td><td><img title="Lorenz gauge" alt="QED-126" src="images/qed/QED-126.gif" align="texttop" vspace="2"></td><tr>

Deletions:
<td valign=top>Lorentz gauge</td><td>  </td><td><img title="Lorentz gauge" alt="QED-126" src="images/qed/QED-126.gif" align="texttop" vspace="2"></td><tr>

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Edited on 2008-02-24 03:25:33 by CharlesFrancis

Additions:

Lemma:  The equal time commutator of charge with the Hamiltonian density is zero,

Deletions:

Lemma:  The equal time commutator of charge with the interaction Hamiltonian is zero,

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Edited on 2008-02-21 23:32:00 by CharlesFrancis

Additions:
using the generalised scaling property of the delta function applied to the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x − y is spacelike.

Deletions:
using the generalised scaling property of the delta function applied to the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

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Edited on 2008-02-14 00:28:10 by CharlesFrancis

Additions:
Proof:  From the above formulae,

using the generalised scaling property of the delta function applied to the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

Deletions:
Proof:   From the above formulae,
since the particle satisfies the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

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Edited on 2008-02-14 00:19:59 by CharlesFrancis

Additions:
since the particle satisfies the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

Deletions:
since the particle satisfies the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

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Edited on 2008-01-26 00:08:00 by CharlesFrancis

Additions:

Lemma:  The equal time commutator of charge with the interaction Hamiltonian is zero,

Deletions:

Lemma:  Charge commutes with the interaction Hamiltonian,

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Edited on 2008-01-16 06:51:53 by CharlesFrancis

Additions:
Proof:  We have,

""<table><td valign=top>Maxwell’s equations</td><td>  </td><td><img title="Maxwell’s Equations" alt="QED-125" src="images/qed/QED-125.gif" align="texttop" vspace="0"></td><tr>

<td valign=top>Lorentz gauge</td><td>  </td><td><img title="Lorentz gauge" alt="QED-126" src="images/qed/QED-126.gif" align="texttop" vspace="2"></td><tr>
These three equations are summarised in Faraday’s law,

Deletions:
Proof;  We have,

""<table><td valign=top>Maxwell’s equations</td><td>  </td><td><img title="Maxwell’s Equations" alt="QED-125" src="images/qed/QED-125.gif" align="texttop" vspace="3"></td>,<tr>

<td valign=top>Lorentz gauge</td><td>  </td><td><img title="Lorentz gauge" alt="QED-126" src="images/qed/QED-126.gif" align="texttop" vspace="1"></td><tr>
These three equations are summarised in Faraday’s law

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Edited on 2008-01-16 06:47:56 by CharlesFrancis

Additions:
,

Maxwell’s equations
Lorentz gauge
Faraday tensor

Deletions:
,

Maxwell’s equations	nbsp;nbsp;
Lorentz gauge	nbsp;nbsp;
Faraday tensor	nbsp;nbsp;

</td></table>""

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Edited on 2008-01-16 06:46:11 by CharlesFrancis

Additions:
,

Maxwell’s equations	nbsp;nbsp;
Lorentz gauge	nbsp;nbsp;
Faraday tensor	nbsp;nbsp;

</td></table> From the definition of the Faraday tensor, <span class=math><i>a</i>, <i>b</i>, <i>c</i> = 0, 2, 3 gives <span class=math><i>a</i>, <i>b</i>, <i>c</i> = 0, 3, 1 gives <span class=math><i>a</i>, <i>b</i>, <i>c</i> = 0, 1, 2 gives <span class=math><i>a</i>, <i>b</i>, <i>c</i> = 1, 2, 3 gives the magnetostatic law, Thus, in a treatment based on <img title="The classical vector field" alt="QED-142" src="images/qed/QED-142.gif" align="texttop" vspace="0">"" the magnetic laws are mathematical identities, not physical laws.

Deletions:

From the definition of the Faraday tensor
a, b, c, = 0, 2, a, 3 gives
a, b, c, = 0, 3, 1 gives
a, b, c, = 0, 1, a, 2 gives
a, b, c, = 1, 2, 3 gives the magnetostatic law,
Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws.

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Edited on 2008-01-16 06:38:08 by CharlesFrancis

Additions:
Proof;  We have,

From the definition of the Faraday tensor

we have the identity,

a, b, c, = 0, 2, a, 3 gives

a, b, c, = 0, 3, 1 gives

a, b, c, = 0, 1, a, 2 gives

a, b, c, = 1, 2, 3 gives the magnetostatic law,

Deletions:
Proof: of Maxwell’s equations

We have,

From

we have the identity

abc = 023 gives

abc = 031 gives

abc = 012 gives

a,b,c = 1,2,3 gives the magnetostatic law

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Edited on 2008-01-16 06:13:56 by CharlesFrancis

Additions:

since the particle satisfies the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

Conserved Current

Theorem:  is a classical conserved current,

Lemma:  Charge commutes with the interaction Hamiltonian,

Proof of Lemma:

Take the Hermitian conjugate and postmultiply by γ⁰,

So, by commuting the terms,

Proof of theorem:  We have, from Ehrenfest’s theorem, that a classical quantity obeys

.

The first term is zero by the lemma. From the solution of the Dirac equation, we have that

Taking the Hermitian conjugate, post multiplying by γ⁰, and using the conjugacy relation, γ⁰γ^a†γ⁰ = γ^a gives

Similarly

Current is given by

Normal ordering transposes the creation and annihilation operators, but not the spin indices. Expanding the creation and annihilation operators in terms of momentum and differentiating the particle term gives,

The other terms are found to be zero in the same way.
Return

Maxwell’s Equation’s

Proof: of Maxwell’s equations
We have,

Hence,

The first term gives the electrostatic equation

The last three terms give

These equations are summarised in the Ampère-Maxwell law

From

we have the identity

abc = 023 gives

abc = 031 gives

abc = 012 gives

These three equations are summarised in Faraday’s law

a,b,c = 1,2,3 gives the magnetostatic law

Thus, in a treatment based on the magnetic laws are mathematical identities, not physical laws.
Return

Deletions:
.

since the particle satisfies the mass shell condition. The integral is Lorentz invariant and is zero when x⁰ − y⁰ = 0. We conclude that it is zero whenever x-y is spacelike.

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Edited on 2008-01-15 05:46:58 by CharlesFrancis

Additions:

Theorem:  The anticommutation relation for the Dirac field and the Dirac adjoint is zero outside the light cone.

Deletions:

Theorem:  The anticommutation relation for the Dirac field and the Dirac adjoint is zero outside the light cone.

Return

""<span class=math><b>Theorem:</b> ; The anticommutation relation for the Dirac field and the Dirac adjoint is zero outside the light cone.

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Edited on 2008-01-15 05:44:50 by CharlesFrancis

Additions:
where ^T denotes that α and β are transposed. Using the resolution of unity and the solution of the Dirac equation,

Deletions:
where ^{T"2 denotes that} <span class=math>α and <span class=math>β"" are transposed. Using the resolution of unity and the solution of the Dirac equation,

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Oldest known version of this page was edited on 2008-01-15 05:42:52 by CharlesFrancis []

Locality of Dirac Field Operators

Proof:  Using the identity (true in a particular basis, so true in any basis),
and (from the solution of the Dirac Equation)
we find that

Similarly,

We have
where ^{T"2 denotes that} <span class=math>α and <span class=math>β are transposed. Using the resolution of unity and the solution of the Dirac equation, <img title="expand in momentum space" alt="QED-21" src="images/qed/QED-21.gif" align="texttop" vspace="2"> <img title="Substitute" alt="QED-22" src="images/qed/QED-22.gif" align="texttop" vspace="2"> Likewise for the antiparticle, <img title="expand in momentum space" alt="QED-23" src="images/qed/QED-23.gif" align="texttop" vspace="2"> <img title="substitute" alt="QED-24" src="images/qed/QED-24.gif" align="texttop" vspace="2"> Substituting <span class=math><b><i>p</i></b> → −<b><i>p</i></b> at <span class=math><i>x</i>⁰ = <i>y</i>⁰ <img title="expand in momentum space" alt="QED-26" src="images/qed/QED-26.gif" align="texttop" vspace="2"> Adding at <span class=math><i>x</i>⁰ = <i>y</i>⁰ gives the equal time anticommutator, <img title="substitute" alt="QED-27" src="images/qed/QED-27.gif" align="texttop" vspace="2"> As required. <a href=http://www.teleconnection.info/rqg/QED#SubReturn1>Return</a>"»"