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markwh04@yahoo.com
science forum Guru Wannabe
Joined: 12 Sep 2005
Posts: 137
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 Posted: Sat May 20, 2006 3:38 pm Post subject:
Fermions as gauge fields
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A well-known problem with the Dirac field is that its Lagrangian is
singular, linear in the velocities, and -- under the action of the
field equations -- actually evaluates to 0. The first two features are
generally the hallmark of a dynamic system when the coordinates and
velocities have been mixed together and haven't yet been separated out.
It's not too difficult to see how the situation arises. The Dirac
equation is a special case of the Dirac-Kemmer equation, which applies
for general spin > 0. The Maxwell system (and more generally, the
Maxwell-Proca system), when reduced to a first order system will yield
the spin 1 version of Dirac-Kemmer.
Therefore, it's natural to regard the Dirac field, analogous to the
Maxwell field as field strengths, themselves derivable from a
potential. Indeed, it is possible to write the Dirac equation in a form
analogous to the Maxwell equations, which bolsters this analogy.
The general solution to the Dirac equation (iD - m) psi = 0, for
non-zero mass, may be arrived at algebraically: psi = (iD + m) chi. The
field chi plays the role of the potential and is only subject to the
Klein-Gordon equation ([] + m^2) chi = 0. From this, the Dirac equation
for the "field strength" follows.
Thus, the Dirac equation on psi may be replaced by the Klein-Gordon
equation on chi.
The gauge invariance, here, is chi -> chi + (iD - m) omega, with omega
also satisfying the Klein-Gordin equation.
A Lagrangian may actually be written down which is closely analogous to
those for the Maxwell and scalar fields:
L = (D chi)-bar (D chi) - m^2 chi-bar
chi.
This corrects the problem with the original Lagrangian: it is
non-singular. In fact, when the conjugate momenta are substituted in,
the Lagrangian takes on the form
L = p-bar p - m^2 chi-bar chi
with p-bar and p respectively conjugate to chi and chi-bar.
In fact, the momenta are linear in D chi, D chi-bar, chi and chi-bar
and psi and psi-bar may be written as linear expressions in the momenta
and gauge fields.
| Quote: | From this, the Hamiltonian may be derived, and commutator relations
written down |
[chi(A), p-bar(B)] = i h-bar (A-bar B)
where the fields are smeared with Grassmann functions
chi(A) = integral A-bar(x) chi(x) d^3 x
p-bar(B) = integral p-bar(x) B(x) d^3 x
A-bar B = integral A-bar(x) B(x) d^3 x.
For the conjugate fields (noting the anticommutativity of the Grassmann
parameters A, B), one gets
[chi-bar(B), p(A)] = -i h-bar (A-bar B).
| Quote: | From this, the commutators relations for the original fields (psi,
psi-bar) may easily be written down. Up to rescaling, it matches the |
usual commutator relations. The required rescaling is psi = (iD + m)
chi (2 m)^{-1/2}, similarly for psi-bar. |
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Cl.Massé
science forum Guru Wannabe
Joined: 24 Mar 2005
Posts: 149
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| Posted: Thu May 25, 2006 1:21 am Post subject:
Re: Fermions as gauge fields
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<markwh04@yahoo.com> a écrit dans le message de news:
1147904774.238751.294500@38g2000cwa.googlegroups.com
| Quote: | The general solution to the Dirac equation (iD - m) psi = 0, for
non-zero mass, may be arrived at algebraically: psi = (iD + m) chi. The
field chi plays the role of the potential and is only subject to the
Klein-Gordon equation ([] + m^2) chi = 0. From this, the Dirac equation
for the "field strength" follows.
Thus, the Dirac equation on psi may be replaced by the Klein-Gordon
equation on chi.
The gauge invariance, here, is chi -> chi + (iD - m) omega, with omega
also satisfying the Klein-Gordin equation.
A Lagrangian may actually be written down which is closely analogous to
those for the Maxwell and scalar fields:
L = (D chi)-bar (D chi) - m^2 chi-bar
chi.
This corrects the problem with the original Lagrangian: it is
non-singular. In fact, when the conjugate momenta are substituted in,
the Lagrangian takes on the form
L = p-bar p - m^2 chi-bar chi
with p-bar and p respectively conjugate to chi and chi-bar.
In fact, the momenta are linear in D chi, D chi-bar, chi and chi-bar
and psi and psi-bar may be written as linear expressions in the momenta
and gauge fields.
From this, the Hamiltonian may be derived, and commutator relations
written down
[chi(A), p-bar(B)] = i h-bar (A-bar B)
where the fields are smeared with Grassmann functions
chi(A) = integral A-bar(x) chi(x) d^3 x
p-bar(B) = integral p-bar(x) B(x) d^3 x
A-bar B = integral A-bar(x) B(x) d^3 x.
For the conjugate fields (noting the anticommutativity of the Grassmann
parameters A, B), one gets
[chi-bar(B), p(A)] = -i h-bar (A-bar B).
From this, the commutators relations for the original fields (psi,
psi-bar) may easily be written down. Up to rescaling, it matches the
usual commutator relations. The required rescaling is psi = (iD + m)
chi (2 m)^{-1/2}, similarly for psi-bar.
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All good. Feynman already derived such a decomposition in his paper about
weak interaction, which is besides the original one of Dirac. One Loren(t)z
also derived an equation for the electromagnetic field similar to the Dirac
one. But what are the consequences?
That a Dirac field is a gauge fields doesn't follow, since a gauge filed is
characterised by the coupling, and not by the equation of motion, even
though the gauge transformation is similar. It compares to the Maxwell
field, but the Dirac equation remains necessary in order to describe a
spin-1/2, that is, the internal geometry.
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability. |
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markwh04@yahoo.com
science forum Guru Wannabe
Joined: 12 Sep 2005
Posts: 137
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| Posted: Thu Jun 08, 2006 8:03 am Post subject:
Re: Fermions as gauge fields
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Cl.Massé wrote:
| Quote: | All good. Feynman already derived such a decomposition in his paper about
weak interaction, which is besides the original one of Dirac.
|
Nobody has successfully done this in the presence of the electroweak
force, as far as I'm aware (present article included).
It is doable for general Yang-Mills forces, but only if the left and
right sectors match.
If it can be done for chiral interactions that may fill in the last
missing piece to rendering the total Lagrangian of the Standard Model
completely in a "sum of squares" form
L = <Z1,Z1> + <Z2,Z2> + ... + <Zn,Zn>
with suitable metrics over each of the sectors 1, 2, ..., n. Such a
Lagrangian factors right down the middle
L = <D,D>
D = Z1 k1 + Z2 k2 + .. + Zn kn
for a suitably-defined algebra.
| Quote: | That a Dirac field is a gauge fields doesn't follow, since a gauge filed is
characterised by the coupling
|
A gauge theory is characterized by the presence of first class
constraints in the Lagrangian (or equivalent, the occurrence of a
0-eigenspace in the mass matrix d^2L/dv^2) -- a situation that applies
to the fermion Lagrangian ordinary used for the Dirac-Kemmer equation
.... which ("which" meaning the Lagrangian) is singular of maximal
order.
| Quote: | It compares to the Maxwell field, but the Dirac equation remains necessary in
order to describe a spin-1/2, that is, the internal geometry.
|
The analogy is quite deep. The same situation exists for the spin 0, 1,
2, ... fields when you start out with the Dirac-Kemmer Lagrangian.
All the usual approaches (Dirac, Maxwell, etc.) apply to all fields of
all spins in all the instances of the generzalized
Poincare'/Galilei/Euclidean group. There is even a Dirac form for the
Galilei tardyon sector, one for the "synchron" sector (the massless
Galilean Wigner class), one for the Poincare' "tachyon" sector (indeed,
it's none other than Dirac with a chiral mass term). This is described
in further detail in an update shortly to be made to "The Wigner
Classification for Galilei/Poincare/Euclid" under
http://federation.g3z.com/Physics/Index.htm.
Also of interest, "The Yang-Mills Equations in Maxwell Form", and "The
Dirac Equation in Maxwell Form", in the latter the analogy drawn deeper
with the field expanded in the analogue of the A and phi potentials;
and the transformation properties and invariants written out explicitly. |
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Cl.Massé
science forum Guru Wannabe
Joined: 24 Mar 2005
Posts: 149
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| Posted: Mon Jun 12, 2006 9:20 pm Post subject:
Re: Fermions as gauge fields
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I wrote:
| Quote: | Feynman already derived such a decomposition in his paper
about weak interaction, which is besides the original one of Dirac.
|
<markwh04@yahoo.com> a écrit dans le message de news:
1149535247.827723.271720@g10g2000cwb.googlegroups.com
| Quote: | Nobody has successfully done this in the presence of the electroweak
force, as far as I'm aware (present article included).
|
Phys. Rev. 109(1958)193.
| Quote: | It is doable for general Yang-Mills forces, but only if the left and
right sectors match.
|
Feynman did it for the V-A interaction, with parity broken.
| Quote: | A gauge theory is characterized by the presence of first class
constraints in the Lagrangian (or equivalent, the occurrence of a
0-eigenspace in the mass matrix d^2L/dv^2)
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Or in the Hamiltonian formalism, by a coupling of a given form. A gauge
interaction is defined by the invariance under a local transformation.
--
~~~~ clmasse on free F-country
Liberty, Equality, Profitability. |
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