Counting degrees of freedom in field theory

In summary, gauge and diffeomorphism invariance play crucial roles in reducing the number of physical degrees of freedom in field theories. These symmetries allow us to remove unphysical degrees of freedom, leaving us with the correct number of degrees of freedom for the physical behavior of the particles. This can be seen in the examples of classical electromagnetism and general relativity, where the number of degrees of freedom is reduced from the naive count due to these symmetries.
  • #1
Frank Castle
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I'm having a bit of trouble with counting the number of physical ("propagating") degrees of freedom (dof) in field theories. In particular I've been looking at general relativity (GR) and classical electromagnetism (EM).

Starting with EM:

Naively, given the 4-potential ##A^{\mu}## has four components one would think there are 4 dof in EM. However, it is known that the photon only has 2 dof, its polarisation states (how is this known? Is it from empirical data, or can it be shown mathematically? Does it follow from Maxwell's equations, i.e. one can only satisfy Maxwell's equations in a consistent manner if the ##\mathbf{E}## and ##\mathbf{B}## fields are both transverse to the direction of propagation (and mutually orthogonal)?! In the case of monochromatic light, say one oscillates horizontally and the other vertically, then an individual photon has two possible polarisation states.)

To solve this discrepancy, we first notice, from studying the Lagrangian for EM (in vacuum) $$\mathcal{L}_{EM}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ that ##A^{0}## has no kinetic term and hence is not dynamical, meaning that if we are given some initial data ##A^{i}## and ##\dot{A}^{i}## at time ##t_{0}##, then ##A^{0}## is fully determined by the equation of motion (eom) ##\nabla\cdot\mathbf{E}=0##. Hence ##A^{0}## is not independent: we don't get to specify it arbitrarily on the initial time slice. As such, this reduces the number of dof to 3.

Up to this point I think I understand the argument, however, I hit a bit of a stumbling block when gauge invariance is taken into account...
... Next noting that ##\mathcal{L}_{EM}## is invariant under a gauge transformation of ##A^{\mu}##, i.e. ##A^{\mu}\rightarrow A^{\mu}+\partial^{\mu}\Lambda(x)##. Hence, we can choose ##\Lambda(x)## such that ##A^{\mu}## satisfies $$\partial_{\mu}A^{\mu}=0$$
Is the point here that this equation places a constraint on the remaining 3 fields ##A^{i}##, such that, given the knowledge of ##A^{0}## (already determined by the initial data and eom ##\nabla\cdot\mathbf{E}=0##), one of the fields ##A^{i}## is fully determined by the other two (and A^{0}). Hence, the gauge symmetry permitting a arbitrary choice of ##\Lambda(x)## enables one to remove another dof, thus leaving 2 physical dof remaining?!Next, considering GR:

Again, starting with the metric tensor ##g_{\mu\nu}(x)##, given that it is symmetric one would naively think that the theory has 10 physical dof. However, if one takes into account the Bianchi identities: $$\nabla_{\mu}G^{\mu\nu}=\nabla_{0}G^{0\nu}+\nabla_{i}G^{i\nu}=0$$ we see that these can be used to remove 4 dof (is the argument here that, given the knowledge of ##G^{0\nu}##, one can fully determine the components ##G^{i\nu}## by noting that ##\nabla_{i}G^{i\nu}=-\nabla_{0}G^{0\nu}##, or vice verse?!). This leaves us with ##10-4=6## dof.
Secondly, we note that GR is diffeomorphism invariant, i.e. the Einstein-Hilbert action $$S_{EH}=\frac{1}{8\pi G}\int d^{4}x\sqrt{-g}\,\mathcal{R}$$ is invariant under diffeomorphisms: $$x^{\mu}\rightarrow x^{\mu}+\xi^{\mu}$$ (where ##\zeta^{\mu}## is a vector field generating the diffeomorphism).

This is where I'm slightly unsure again...
... I know that under a diffeomorphism the metric transforms (to first order) as $$g'_{\mu\nu}(x')=\frac{\partial x^{\alpha}}{\partial x'^{\mu}}\frac{\partial x^{\beta}}{\partial x'^{\nu}}g_{\alpha\beta}(x)=g_{\mu\nu}(x)-g_{\mu\beta}(x)\partial_{\nu}\xi^{\beta}-g_{\alpha\nu}(x)\partial_{\mu}\xi^{\alpha}$$

Is the point that we are free to choose the 4 components ##\xi^{\mu}## such that ##g_{\mu\nu}(x)## satisfies a particular constraint (say ##\nabla^{\mu}g_{\mu\nu}(x)=\nabla^{0}g_{0\nu}(x)=\nabla^{i}g_{i\nu}(x)=0##), thus removing another 4 dof?!

Hence, we find that GR has ##10-4-4=2## physical dof, which are the polarisation states of the graviton.

Hopefully the questions I have are clear in the text above; I've tried to highlight each one by putting them in italics. The basic summary is how does one use gauge (or diffeomorphism) invariance to remove dof, and are they unphysical dof precisely because one can remove them by making a gauge transformation?
 
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  • #2

Thank you for your question about counting the number of physical degrees of freedom in field theories. I will try my best to answer your questions and clarify any confusion you may have.

First, regarding the number of degrees of freedom in classical electromagnetism (EM), you are correct in noting that there are only two physical degrees of freedom, corresponding to the two polarization states of the photon. This is known both empirically, through experimental observations of the behavior of light, and mathematically, through the equations of motion for the EM field. The fact that the electric and magnetic fields must be transverse to the direction of propagation is a consequence of Maxwell's equations and is related to the fact that there are only two independent polarization states.

Moving on to the role of gauge invariance in reducing the number of degrees of freedom in EM, your understanding is correct. The gauge transformation allows us to remove one degree of freedom from the vector potential, leaving us with three independent degrees of freedom. This can also be seen through the constraint equation you mentioned, which relates the electric field to the divergence of the vector potential. This further reduces the number of degrees of freedom to two, matching what we expect from the physical behavior of the photon.

Next, for general relativity (GR), you are correct in noting that the metric tensor has ten components, but not all of them are independent. The Bianchi identities allow us to remove four degrees of freedom, leaving us with six. This is because the Bianchi identities relate the time and spatial components of the Einstein tensor, so once we know one set, the other is determined. This leaves us with six independent components of the metric tensor.

Diffeomorphism invariance, as you correctly point out, also plays a role in reducing the number of physical degrees of freedom in GR. The freedom to perform diffeomorphisms means we can choose coordinates in a way that satisfies certain constraints, such as the ones you mentioned. This further reduces the number of independent degrees of freedom to two, corresponding to the two polarization states of the graviton.

To summarize, gauge and diffeomorphism invariance play crucial roles in reducing the number of physical degrees of freedom in field theories. These symmetries allow us to remove unphysical degrees of freedom, leaving us with the correct number of degrees of freedom for the physical behavior of the particles. I hope this has helped to clarify your understanding of counting degrees of freedom in
 

FAQ: Counting degrees of freedom in field theory

What is the concept of degrees of freedom in field theory?

The concept of degrees of freedom in field theory refers to the number of independent variables or parameters that can vary in a physical system. In other words, it represents the number of ways in which a system can move or evolve.

How do you count the degrees of freedom in a field theory?

To count the degrees of freedom in a field theory, one needs to identify all the independent variables or parameters in the system and determine how many of them can vary independently. This can be done by considering the number of independent positions and momenta in the system.

What is the significance of counting degrees of freedom in field theory?

Counting degrees of freedom in field theory is important because it helps us understand the behavior and dynamics of a physical system. It also allows us to predict the possible outcomes and constraints of a system based on the number of degrees of freedom.

Can the number of degrees of freedom change in a physical system?

Yes, the number of degrees of freedom in a physical system can change. This can happen when the system undergoes a phase transition, such as a change in temperature or pressure, which can alter the number of independent variables or parameters in the system.

How does the concept of degrees of freedom relate to entropy in field theory?

The concept of degrees of freedom is closely related to entropy in field theory. Entropy is a measure of the disorder or randomness of a system, and it is directly proportional to the number of degrees of freedom in the system. As the number of degrees of freedom increases, so does the entropy of the system.

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