I have been studying quantum field theory and I am currently in the Lagrangian field theory chapter in my book. Now it says that the energy momentum tensor is as follows:(adsbygoogle = window.adsbygoogle || []).push({});

T^{μ}_{ν}= [∂L/∂(∂_{μ}φ) * ∂_{ν}φ] - δ^{μ}_{ν}L

Note: I am using L to symbolize Lagrangian density and not just Lagrangian since the latex box doesn't have the curly L in it. Those two indices on the term on the right go to the Kronecker delta, not to the L.

It follows up by saying that

T^{0}_{0}= [∂L/∂φ' * φ'] - L (in the book, they have φ with a dot over it instead of φ' )

Now I just want to verify some things:

Does φ with a dot over it (or φ' ) refer to ∂φ/∂t ?

If so, then this would be the velocity of the field correct, since it is the first time derivative? I ask this because unlike in the classical mechanics example problems that the book gave (where I was just dealing with position functions of time), fields φ(x,t) are functions of both time and space.

Also, how does a field itself have a velocity if it permeates all of space? I could see how the particles that are generated from fluctuations of said field have a velocity. How does the field itself have a velocity? Does velocity for a field refer to how fast the particles generated from that field move or how fast the field's fluctuations are? Is it something else?

Finally, how can a field itself have mass? If you plug a Lagrangian density into the Euler Lagrange equations for some given Lagrangian densities, then you sometimes get mass terms in the equations of motion that you derive? Once again, I see how a particle that comes from a field has mass, but I don't see how the field itself has mass (example, I see how the gluon has mass, but how does the strong field have mass?)

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# Properties of fields in quantum field theory

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