Properties of fields in quantum field theory

[space-time]

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:

T^μ_ν= [∂L/∂(∂_μφ) * ∂_νφ] - δ^μ_νL

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

T⁰₀ = [∂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?)

 

[snatchingthepi]

As to the last question about the fields having mass, I would suppose (if I am incorrect please correct me) that since the fields have energy, they must also have a mass equivalent to that energy.

 

[bhobba]

As to the last question about the fields having mass, I would suppose (if I am incorrect please correct me) that since the fields have energy, they must also have a mass equivalent to that energy.

That's not true. Its a VERY VERY common misconception and anyone that believes can be excused. E=MC^2 says mass is a form of energy. Energy comes in a lot of forms eg EM field energy - mas is simply one of those forms. It does not imply energy is a form of mass.

Thanks
Bill

 

[snatchingthepi]

That's not true. Its a VERY VERY common misconception an anyone that believes can be excused. E=MC^2 says mass is a form of energy. Energy comes in a lot of forms eg EM field energy - mas is simply one of those forms. It does not imply energy is a form of mass.

Thanks
Bill

May you please PM some of the details, or refer me to a section in a book or even a Wikipedia article so that I can learn more about what you're saying?

 

[bhobba]

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

Yes

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.

No. Its the rate of change of the field value at a specific point. Think of a stretched rope with waves - the derivative gives a rate of change of the height of a little element of the rope - that's not the velocity of the waves of the rope.

Also, how does a field itself have a velocity if it permeates all of space?

The same way waves in a rope have a velocity.

Finally, how can a field itself have mass?

In QFT they can - but I will let someone more conversant in that subject explain why. In classical field theory the field doesn't have mass - it has energy via Noethers theorem and the modern definition of energy - but not mass.

Thanks
Bill

 

[bhobba]

May you please PM some of the details, or refer me to a section in a book or even a Wikipedia article so that I can learn more about what you're saying?

Its simple logic.

The modern definition of energy is via Noether's beautiful and deep theorem:
http://www.sjsu.edu/faculty/watkins/noetherth.htm

Although you may not have seen it done this way the correct derivation of E=MC^2 uses that:
http://fma.if.usp.br/~amsilva/Livros/Zwiebach/chapter5.pdf

It proves the energy of a free relativistic particle at rest is MC^2. It does not prove energy in a general sense has mass - indeed from the modern definition based on Noethers theorem it makes no sense.

The reason its a very common error is people are often very loose about reasoning with it - even some experts who should know better.

However if you want to pursue it start a thread in our relativity sub-forum.

Added Later

The following may also help:
http://www.quora.com/Does-energy-have-mass

Thanks
Bill