Most recent edit on 2008-08-13 03:51:13 by CharlesFrancis

Additions:

← Feynman Diagrams ↑ →

The accepted wisdom is that Feynman diagrams should not be taken literally, but should merely be treated as aids to calculation. That is not how Feynman regarded them, as described in, for example, QED: The Strange Theory of Light and Matter». In the interpretation of quantum mechanics used here, they are understood as statements in quantum logic. Since we cannot say what combination of particle interactions takes place between measurements, we must sum the possibilities using quantum logical OR. In Feynman diagrams only lines and vertices have meaning. They represent structures of matter as configurations of particle interactions in the absence of background of space or spacetime.

Deletions:

← Feynman Diagrams →

Edited on 2008-06-29 03:10:48 by CharlesFrancis

Additions:
""<table><td valign=top> To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by <a href=http://en.wikipedia.org/wiki/Graph_(mathematics)>graphs</a>»». A particle created at x0 may be annihilated at a later time y0. An antiparticle created at x0 may be annihilated at an earlier time y0. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering.</td><td><img class=right title="Contraction represented by lines" alt="Feynman-21" src="images/feynman/Feynman-21N.gif" ></td></table>

Deletions:
""<table><td valign=top> To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by graphs». A particle created at x0 may be annihilated at a later time y0. An antiparticle created at x0 may be annihilated at an earlier time y0. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering.</td><td><img class=right title="Contraction represented by lines" alt="Feynman-21" src="images/feynman/Feynman-21N.gif" ></td></table>

Edited on 2008-06-29 03:09:00 by CharlesFrancis

Additions:
""<table><td valign=top> To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by graphs». A particle created at x0 may be annihilated at a later time y0. An antiparticle created at x0 may be annihilated at an earlier time y0. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering.</td><td><img class=right title="Contraction represented by lines" alt="Feynman-21" src="images/feynman/Feynman-21N.gif" ></td></table>

Deletions:
""<table><td valign=top> To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by graphs. A particle created at x0 may be annihilated at a later time y0. An antiparticle created at x0 may be annihilated at an earlier time y0. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering.</td><td><img class=right title="Contraction represented by lines" alt="Feynman-21" src="images/feynman/Feynman-21N.gif" ></td></table>

Edited on 2008-05-26 11:30:04 by CharlesFrancis

Additions:
The perturbation expansion

Deletions:
The perturbation expansion

Edited on 2008-05-26 10:27:27 by CharlesFrancis

No differences.

Edited on 2008-05-26 10:26:26 by CharlesFrancis

Additions:

Wick’s Theorem

Deletions:

Wick’s Theorem

Regularisation

A detailed treatment of regularisation goes beyond the scope of my immediate objectives for this site. Causal perturbation theory», described in Finite Quantum Electrodynamics» by G. Scharf», shows how loop diagrams can be calculated rigorously using the method of Epstein and Glaser. The step functions, Θ, are replaced with a smooth», monotonic» function, χ, and the advanced and retarded parts are calculated separately. Then a limit is taken in which χ becomes a step function and the advanced and retarded parts are described as distributions». When the distributions are combined the integrations are finite and the usual Feynman rules are obtained for diagrams without loops.
More ad hoc treatments can be justified by observing that integrals always stand for finite sums. The step functions can be replaced with Feynman-41

and

, for some small positive time interval χ. This introduces an upper bound, or cut-off, in energy. Then, when the integrals are evaluated, cut-off dependent terms are discarded, and indeterminacy in the result is removed by fixing charge and mass to be the measured values.
The problem arises because quantum theory is conventionally set up as a continuum theory, using an infinite dimensional Hilbert space. In such a theory Feynman-43

is a divergent quantity and has no meaning. Mathematically the continuum only exists as a result of infinite limiting procedures, and strictly refers to a sequence indexed by n, such that taking n larger than some value, N, makes no practical difference to results. On this site, Hilbert space is regarded as strictly N-dimensional, for a large value of N. Feynman-44

is well defined, but in terms of creation and annihilation operators, Feynman-45

would mean that a particle is annihilated at the moment of its creation. Physically this is not sensible for a theory of particles. Regularisation simply means that amplitudes from this non-physical process must be excluded from Feynman diagrams and the dependency on N is removed. This analysis of the origin of ultraviolet divergences is essentially the same as that given in causal perturbation theory». The difference is that here “switching off and switching on” of the interaction at Feynman-46

is a physical constraint, and means that only one interaction takes place for each particle in any instant, or equivalently that a particle cannot interact again at the moment of its creation.

Renormalisation

[Renormalisation is] “just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number.” — Paul Dirac.

“The shell game that we play ... is technically called ‘renormalization’. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” — Richard Feynman.

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’s Equations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. Values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.
Feynman-67

Edited on 2008-05-20 03:53:18 by CharlesFrancis

Additions:

“The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” — Richard Feynman.

Edited on 2008-05-20 03:50:01 by CharlesFrancis

Additions:
The derivations of Maxwell’s Equations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. Values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:
The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-05-20 03:46:30 by CharlesFrancis

No differences.

Edited on 2008-05-20 03:45:30 by CharlesFrancis

Additions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-05-20 03:38:48 by CharlesFrancis

Additions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-05-20 03:33:27 by CharlesFrancis

Additions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-05-20 03:30:35 by CharlesFrancis

Additions:
“The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” — Richard Feynman, Nobel laureate 1965.

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:
“The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” — Richard Feynman, Nobel laureate 1965.>><img title="virtual photon cloud?" class=left alt="Feynman-66" src="images/feynman/Feynman-66N.gif">Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by [[http://en.wikipedia.org/wiki/Renormalization renormalisation]]. Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of <a href=http://www.teleconnection.info/rqg/CEM#Maxwell’s» equations>Maxwell’sEquations</a> and the <a href=http://www.teleconnection.info/rqg/CEM#TheLorentzForceLaw>Lorentz» Force law</a> show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined <a href=http://www.teleconnection.info/rqg/Feynman#Regularisation>above</a>""», it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-05-20 03:29:11 by CharlesFrancis

Additions:

“[Renormalisation is] just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number.” — Paul Dirac, Nobel laureate 1933.

“The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.” — Richard Feynman, Nobel laureate 1965.>><img title="virtual photon cloud?" class=left alt="Feynman-66" src="images/feynman/Feynman-66N.gif">Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by [[http://en.wikipedia.org/wiki/Renormalization renormalisation]]. Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of <a href=http://www.teleconnection.info/rqg/CEM#Maxwell’s» equations>Maxwell’sEquations</a> and the <a href=http://www.teleconnection.info/rqg/CEM#TheLorentzForceLaw>Lorentz» Force law</a> show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined <a href=http://www.teleconnection.info/rqg/Feynman#Regularisation>above</a>""», it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.
Feynman-67

Deletions:
Feynman-66

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.
Feynman-67

Edited on 2008-05-05 01:26:11 by CharlesFrancis

Additions:
The accepted wisdom is that Feynman diagrams should not be taken literally, but should merely be treated as aids to calculation. That is not how Feynman regarded them, as described in, for example, QED: The Strange Theory of Light and Matter». In the interpretation of quantum mechanics used here, they are understood as statements in quantum logic. Since we cannot say what combination of particle interactions takes place between measurements, we must sum the possibilities using quantum logical OR. In Feynman diagrams only lines and vertices have meaning. They represent structures of matter as configurations of particle interactions in the absence of background of space or spacetime.
Feynman-2

where photon creation and annihilation operators are distinguished by the vector index, a. I(x) is the sum of eight terms, each of which can be represented can be represented diagrammatically as a time ordered vertex» or node. Lines above the node correspond to creation operators, and those below the node correspond to annihilation operators. The photon is represented by a wavy line, electrons by a upward arrow and positrons by a downward arrow.

To keep track of the contractions in normal ordering the perturbation expansion, the terms are represented by graphs. A particle created at x⁰ may be annihilated at a later time y⁰. An antiparticle created at x⁰ may be annihilated at an earlier time y⁰. Each contraction is represented by connecting the corresponding lines between vertices, and, at the same time, removing time ordering.

After carrying out the contractions, all topoligically equivalent time ordered diagrams are combined into a single diagram with no time ordering between the nodes. There are n! diagrams with n nodes. So, removing the ordering of nodes generates a factor n! and cancels the factor 1 ⁄ n! in the perturbation expanson, leaving an integration for a diagram with n vertices,

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.
Feynman-67

Vacuum fluctuation diagrams contain no external lines. For any connected Feynman diagram one can draw another diagram which is the same but for the inclusion of a vacuum fluctuation diagram. The effect of adding a vacuum fluctuation diagram to all diagrams is equivalent to multiplying the theory by a constant, and has no impact on the predictions of the theory. Vacuum fluctuation diagrams should therefore be ignored. It is not possible to say that the vacuum is in a state of constant activity, with the spontaneous creation annihilation of virtual particles, because any such particles do not interact with the observable matter of the universe, and should not be regarded as a part of physical reality.

Deletions:
The accepted wisdom is that Feynman diagrams should not be taken literally, but should merely be treated as aids to calculation. That is not how Feynman regarded them, as described in, for example, QED: The Strange Theory of Light and Matter». In the interpretation of quantum mechanics used here, they are understood as statements in quantum logic. Since we cannot say what combination of particle interactions takes place between measurements, we must sum the possibilities using quantum logical OR. In Feynman diagrams only lines and vertices have meaning. They represent structures of matter as configurations of particle interactions in the absence of any background of space or spacetime.
Feynman-2

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.
Feynman-67

Edited on 2008-03-18 01:27:01 by CharlesFrancis

Deletions:
Feynman Diagrams ↑ Scattering →. If |p| > 0 then the photon propagator can be replaced with

Edited on 2008-03-11 03:36:09 by CharlesFrancis

Additions:

← Feynman Diagrams →

Feynman Diagrams ↑ Scattering →. If |p| > 0 then the photon propagator can be replaced with

Deletions:

← Feynman Diagrams →

Edited on 2008-03-11 03:24:33 by CharlesFrancis

Additions:
Feynman-66

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. This is wrong. The derivations of Maxwell’sEquations and the Lorentz Force law show that the observable charge and mass are equal to the bare values. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Deletions:
Feynman-66

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Edited on 2008-02-29 23:21:20 by CharlesFrancis

Additions:
Wick’s theorem» can be used to replace the time ordered product with a normal ordered product by (anti)commuting annihilation operators to the right and creation operators to the left. Let φ be the field operator,

Deletions:
Wick’s theorem» can be used to replace the time ordered product with a normal ordered product by commuting annihilation operators to the right and creation operators to the left. Let φ be the field operator,

Oldest known version of this page was edited on 2008-02-22 09:51:30 by CharlesFrancis []

Page view:

← Feynman Diagrams →

The Time Ordered Vertex for QED

The interaction density for qed is

Wick’s Theorem

The perturbation expansion

is a sum of terms representing n interactions. In quantum logic a sum stands for disjunction. So, the meaning of the perturbation expansion is that we cannot say how many interactions take place in any given physical process.

Wick’s theorem» can be used to replace the time ordered product with a normal ordered product by commuting annihilation operators to the right and creation operators to the left. Let φ be the field operator,

If x is before y, x⁰ < y⁰, the Feynman propagator», D(x − y), gives the amplitude for the creation of an antiparticle at x and its annihilation at y. If x is after y, x⁰ > y⁰ it gives the amplitude for creation of a particle at y and its annihilation at x.

Definition: The Feynman propagator, or contraction of φ^†(y) and φ(x) is

where plus is used for Bosons and minus for Fermions, and Θ is the step function,

Θ(x) = 0 if x ≤ 0,
Θ(x) = 1 if x > 0.

Wick’s Theorem:
For two field operators,

For n field operators,

where 1 ≤ i, j, k ≤ n and contracted pairs are omitted in the normal ordered product.

Proof:

where plus is used for Bosons and minus for Fermions. This is Wick’s theorem for two field operators.
A detailed proof by induction can be carried out for n field operators, but the proof is no more evident than the theorem itself, which just means that we do the normal ordering by carrying out the contractions.

The S-matrix

Initial and final states can be expressed as sums of plane wave states by using the resolution of unity in momentum space. The time evolution between t₀ and t₁ is given by a matrix in momentum space

In the case of scattering, the initial state (generated by a particle accelerator»), and the final state (typically measured by bubble chamber», wire chamber» or silicon detector» are well represented as pure momentum states. In this case the interesting interaction takes place at the scattering event, and t₀ and t₁ are not important.

Definition: The S-matrix^» (or scattering matrix) is

The S-matrix is found from the perturbation expansion by first normal ordering the terms using Wick’s theorem. Then, for the interaction density at x, the creation operator acting on the initial state Feynman-13

gives, for a photon,

for a Dirac particle,

and for an antiparticle,

Similarly, the annihilation operators in the interaction density acting on the final state Feynman-17

gives, for a photon,

for a Dirac particle,

and for an antiparticle,

The Feynman Propagator

Provided that the integrals converge as Feynman-24

, the photon propagator can be evaluated,

where

is a dummy variable, Feynman-28

is a non-vector and Feynman-29

is 3-momentum from j to i; Feynman-30

and

. g is the metric tensor. Similarly, provided that the integrals converge as Feynman-34

, the propagator for a Dirac particle can be evaluated,

These results are demonstrated by combining the integrations for the advanced ( Feynman-37

) and retarded ( Feynman-38

) parts. In practice, we do not have convergence at Feynman-39

in certain diagrams with internals loops. Treating the resulting divergence is regularisation» and can be carried out in a number of ways.

Regularisation

A detailed treatment of regularisation goes beyond the scope of my immediate objectives for this site. Causal perturbation theory», described in Finite Quantum Electrodynamics» by G. Scharf», shows how loop diagrams can be calculated rigorously using the method of Epstein and Glaser. The step functions, Θ, are replaced with a smooth», monotonic» function, χ, and the advanced and retarded parts are calculated separately. Then a limit is taken in which χ becomes a step function and the advanced and retarded parts are described as distributions». When the distributions are combined the integrations are finite and the usual Feynman rules are obtained for diagrams without loops.

More ad hoc treatments can be justified by observing that integrals always stand for finite sums. The step functions can be replaced with Feynman-41

and

, for some small positive time interval χ. This introduces an upper bound, or cut-off, in energy. Then, when the integrals are evaluated, cut-off dependent terms are discarded, and indeterminacy in the result is removed by fixing charge and mass to be the measured values.

The problem arises because quantum theory is conventionally set up as a continuum theory, using an infinite dimensional Hilbert space. In such a theory Feynman-43

is well defined, but in terms of creation and annihilation operators, Feynman-45

is a physical constraint, and means that only one interaction takes place for each particle in any instant, or equivalently that a particle cannot interact again at the moment of its creation.

Conservation of Energy and Momentum

Gather all the exponential terms from internal and external lines with x_i in the exponent. Provided the time from t₀ to t₁ is large, the result is a representation of a delta function

where

refer to the arrowed line coming from vertex, the arrowed line going into the vertex, and the photon line. The delta function shows that the tilda’d quantities are conserved. For internal lines, Feynman-50

, are the dummy variables introduced in the contour integration. For external lines Feynman-51

. Energy, p⁰, was originally defined to be the zero component of a vector. This is not a conserved quantity. Vectors are products of measurement, and only have real meaning in measurement. By definition, internal lines do not correspond to measured states. So, p⁰ has no meaning on internal lines in a Feynman diagram. The conserved tilda’d quantities are of more interest than vector quantities and it is usual to redefine energy.

Redefinition: Energy is the conserved quantity, Feynman-52

, which appears on the lines of a Feynman diagram.

With this definition, energy-momentum, Feynman-53

, is conserved, but is not a vector, and does not obey the mass shell condition on internal lines in Feynman diagrams. Particles are said to be off shell» on internal lines. On external lines, representing measured states, this definition of energy coincides with the original definition for measured states, as the time component of a vector. Particles are said to be on shell» on external lines, meaning that the mass shell condition is obeyed.

Feynman Rules

After using the delta functions to carry out the integrals over tilda’d quantities, and imposing the rule that energy-momentum is conserved at each vertex, there remains an integral for each independent internal loop,

Each vertex contributes a factor

For external lines in the initial state we have, for a photon,

for a Dirac particle,

and for an antiparticle,

For external lines in the final state we have, for a photon,

for a Dirac particle,

and for an antiparticle,

For internal arrowed lines we have

and for internal photon lines we have

In addition there is a minus sign if an odd number of commutations of Fermion creation and annilation operators is required to put the diagram into normal order. The limit ε → 0 should be taken after evaluation of integrals for loops and for the initial and final states. If |p| > 0 then the photon propagator can be replaced with

Certain diagrams contain a divergence when photon energy goes to zero. In this case ε² should be retained until after evaluation of the integral to control the infrared divergence (ε² plays the role of the small photon mass commonly used for this purpose).

Renormalisation

Regularisation removes divergent quantities, but can leave a degree of indeterminism in the result, which is removed by renormalisation». Renormalisation is often justified with an argument that the bare charge appearing in the interaction density, and the bare mass appearing in the Klein-Gordon and Dirac equations are not equal to the observable charge and mass. It is said, for example, that the mass of an electron is modified because it is surrounded by a cloud of “virtual” photons. In fact conservation of energy-momentum implies that the total energy and total momentum of a particle and a cloud of virtual particles in an external line is equal to the energy and momentum of a single free electron in an initial or final state. As described above, values after regularisation are determined by requiring that mass, m, and charge, e, are the observable quantities, but one does not need to use a renormalisation argument to justify this procedure. As outlined above, it is shown in causal perturbation theory» that, when the order of taking limits is correctly followed, the integrations are finite and the contribution of loops on external lines is zero.

Feynman-67

RQG : FeynmanDiagrams

← Feynman Diagrams ↑ →

← Feynman Diagrams →

Wick’s Theorem

Wick’s Theorem

Regularisation

Renormalisation

← Feynman Diagrams →

← Feynman Diagrams →

← Feynman Diagrams →

The Time Ordered Vertex for QED

Wick’s Theorem

The S-matrix

The Feynman Propagator

Regularisation

Conservation of Energy and Momentum

Feynman Rules

Renormalisation