Gauge symmetry
Hello Lawrence:
The post deals with gauge symmetry issues.
My proposal breaks U(1) gauge symmetry. Let's be clear for readers what that means. This is the transformation we have all seen before:
A^{\mu} \rightarrow (\phi,\vector{A})'=(\phi-\frac{\partial \Lambda}{\partial t},A+\nabla \Lambda)
The antisymmetric field strength tensor \nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu} can be represented by the fields E and B defined as follows:
E=-\frac{\partial A}{\partial t}-\nabla \phi
B=\nabla \times A
Plug in the U(1) gauge transformation into those definitions:
E \rightarrow E' = -\frac{\partial A}{\partial t}-\frac{\partial \nabla \Lambda}{\partial t}-\nabla \phi+\nabla \frac{\partial \Lambda}{\partial t}=E
B \rightarrow B'=\nabla \times A+\nabla \times \nabla \Lambda=B
For the E field, the mixed time/space derivatives cancel. For the B field, the curl of curl of a scalar is zero.
The GEM proposal has exactly these two fields E and B. But there are also fields to represent the symmetric tensor. I call them small e and small b, the symmetric analogues to EM's big E and big B. There is also a field for the four along the diagonal. Here are the definitions for the 5 fields in the GEM field strength tensor:
E=-\frac{\partial A}{\partial t}-\nabla \phi
B=\nabla \times A
e=\frac{\partial A}{\partial t}-\nabla \phi-\Gamma_{\sigma}{}^{0u}A^{\sigma}
b=(-\frac{\partial A_{z}}{\partial y}-\frac{\partial A_{y}}{\partial z}-\Gamma_{\sigma}{}^{yz}A^{\sigma},<br />
-\frac{\partial A_{x}}{\partial z}-\frac{\partial A_{z}}{\partial x}-\Gamma_{\sigma}{}^{xz}A^{\sigma},<br />
-\frac{\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y}-\Gamma_{\sigma}{}^{xy}A^{\sigma})
g=(\frac{\partial \phi}{\partial t}-\Gamma_{\sigma}{}^{tt}A^{\sigma}, -\frac{\partial A_{x}}{\partial x}-\Gamma_{\sigma}{}^{xx}A^{\sigma}, -\frac{\partial A_{y}}{\partial y}-\Gamma_{\sigma}{}^{yy}A^{\sigma}, -\frac{\partial A_{z}}{\partial z}-\Gamma_{\sigma}{}^{zz}A^{\sigma})
Apply the U(1) gauge symmetry, and it becomes apparent that the E and B fields are fine, but the fields I think deal with gravity, g, e, and b, are not. Gravity and breaking gauge symmetry are linked in the GEM proposal.
Gauge theory is very powerful. Starting from the U(1) symmetry in 4D, people good at this sort of thing can derive the Maxwell equations. That is a reason why if one states their proposal breaks U(1) gauge symmetry, it is reasonable to think the theory cannot regenerate the Maxwell equations. I am trying to do something more, to fundamentally include mass.
Look at one limitation of gauge theories. Let me quote extensively from Michio Kaku's "Quantum Field Theory: A Modern Introduction" p. 106:
Because of gauge invariance, there are also complications when we quantize the theory. A naive quantization of the Maxwell theory fails for a simple reason: the propagator does not exist. To see this let us write down the action in the following form:
\mathcal{L}=1/2 A^{\mu}P_{\mu \nu}\partial^{2}A^{\nu}
where:
P_{\mu \nu}=g_{\mu \nu}-\partial_{\mu}\partial_{\nu}/(\partial)^2
The problem with this operator is that it is not invertible, and hence we cannot construct a propagator for the theory. In fact, this is typical of any gauge theory, not just Maxwell's theory. This also occurs in general relativity and in superstring theory. The origin of the noninvertibility of this operator is because P_{\mu \nu} is a projection operator, that is, its square is equal to itself:
P_{\mu \nu}P^{\nu \lambda}=P_{\mu}^{\lambda}
and it projects out longitudinal states:
\partial^{\mu}P_{\mu \nu}=0
The fact that P_{\mu \nu} is a projection operator, of course goes to the heart of why Maxwell's theory is a gauge theory. This projection operator projects out any states with the form \partial_{\mu}\Lambda, which is just the statement of gauge invariance.
Physicists understand exactly how to deal with this issue: pick a gauge. With the GEM proposal, this choice is not available. That may be a good thing for quantizing the theory.
There is the problem of mass in the Standard Model. The symmetry U(1) \times SU(2)\times SU(3) justifies the number of particles needed for EM (one photon for U(1), the weak force (three W+, W-, and Z for SU(2)), and the strong force (8 gluons for SU(3)). Straight out of the box, the Standard Model works only if all the masses of particles are zero. Something else is needed to break the symmetry. Readers here know the standard answer: the Higgs mechanism uses spontaneous symmetry breaking to introduce mass into the standard model. As far as I know, there is no compelling connection between the Higgs and the graviton.
Let's think on physical grounds about how mass and charge relate to each other. Consider a pair of electrons and a pair of protons, each held 1 cm apart from each other. Release them, and the electrons repel each other, as do the protons. Measure the acceleration. The electrons accelerate more for two distinct reasons. First, there is the difference in inertial mass because an electron weighs 1800x less than a proton, good old F=mA. Second, the gravitational masses will change the total net force, more attraction for the heavier protons, good old F=-Gmm/R^{2}, which would be too subtle to measure directly. One could say that both inertial and gravitational mass break the symmetry of the standard model. In the GEM proposal, the 3 fields (10 total components) of g, e, and b make up the symmetric field strength tensor \nabla^{\mu}A^{\nu}+\nabla^{\nu}A^{\mu} that could do the work of the graviton, while the trace of that matrix could do the work of the Higgs. I am no where near good enough to make those connections solid. I am just pointing out what looks like a duck might be a duck.
Lawrence has pointed out several ways to be a good gauge theory proposal, but I think GEM proposal is heading a different direction. There is a need to break gauge symmetry in a way consistent with gravity and quantum field theory.
doug
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