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 TheCow
Registered: 10/28/02
Posts: 4,637
Last seen: 2 months, 1 day
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 Quantum Field Theory question
#8398242 - 05/13/08 08:08 PM (3 months, 24 days ago) |
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hey, I was just wondering if anyone here whose actually taken graduate level qft or learned it can answer a question I have. Im reading a book which is basically a short (150 pages) intro to quantum field theory. Once Im done with this Im going to move onto a real book, at the same time Im reading "Lie groups, Lie algebras, and some of their applications" for some preparation. Anyways this book tells us that the zero point energy field is infinite, but in all meaningful calculations where we take the difference in energy it cancels out. This seems odd to me, so I figured we could flex the ol latex capabilities now.
So as an example this book shows us a Hamiltonian with an interacting hamiltonian. Basically the interacting hamiltonian is for QED so its the coupling between a radiation field and the electron-positron field.
So the energy eigenstate for the non-perturbed hamiltonian will be
Next we get the first order energy with the perturbation factored in, which cancels out for reasons there are no reason to explain
Finally the second order energy eigenstate is
where  is generally written as <0,H,I> but for some reason thats not working in the latex code.
So anyways this term equals 
So therefore the book claims that now all is well because we have gotten rid of the infinite energy.
They have to preface this by saying that unlike in General Relativity the absolute value of the energy is not meaningful. Eh, I find this to be very strange. In fact the way that the field in general is arrived at seems dubious to me. Can someone explain why we can have the field as an infinite quantization of the harmonic oscillator?
Im sure all this is explained in a better book, but this is all I have at the moment. I think im gonna trade this in at the library for something better.
Ill explain this better if need be, Im a bit saucy at the moment
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 deimya
Phase transiting
Registered: 08/26/04
Posts: 380
Loc: ausländer.ch
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Re: Quantum Field Theory question [Re: TheCow]
#8400310 - 05/14/08 11:01 AM (3 months, 23 days ago) |
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Basically you have it there, any kind of dynamics is a transition from one state of the field to another and every transition probability is strictly proportional to energy differences. It is easy to see in the Heisenberg equations of motion for an operator  :
and there you see that this zero-point energy gets substracted in the commutator. Therefore the dynamics is unaffected by it.
Also, concerning your question, it indeed is a bit dubious how we build a quantum field. The foundations of QFT are still part of a big, open problem in physics, for which you can get 1M$ from the Clay Institute, but it works very well so people are confident. This foundational problem arises in part because of the correspondence principle. That is to build a quantum field theory, you can do so using path integrals or canonical quantization. Most probably you're first being taught canonical quantization in which basically you start with a Hamiltonian or Lagrangian of a classical field and you quantize by replacing the field variables with elements of an algebra obeying some proper commutation (bosonic) or anti-commutation (fermionic) relations, and then finding a suitable realization of these relations as operators acting on a Fock space. Path integrals are not really more rigorous because we don't know how to properly define the set of paths to integrate on.
For canonical quantization, the whole procedure is fishy right from the start because you go from commuting quantity to anti-commuting quantities, so in which initial order do you replace your symbols ? It is at best ambiguous. In general people choose normal ordering, in which all annihilation operators appears to the right of creation operators, but it could be otherwise and then you would arrive at a slightly different Hamiltonian or Lagrangian. This is related to the "Correspondence Problem", but I don't have the mathematical background to really dig meaningfully into this.
Quantum field theories are mathematical messes for which we somehow have no idea how to construct from bottom-down. And this gave rise to more serious infinities than the apparent infinite zero-point energy. These more serious infinities appears directly into correlation function, which are the important quantities relevant for measurements. These infinities are dealt with using the whole renormalization scheme. Basically you end up to trading some predictability power to get predictability at all. This is where these numerical parameters we have to measure to put in the Standard Model come from.
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TheCow
Registered: 10/28/02
Posts: 4,637
Last seen: 2 months, 1 day
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Re: Quantum Field Theory question [Re: deimya]
#8402187 - 05/14/08 07:21 PM (3 months, 23 days ago) |
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Quote:
deimya said:
For canonical quantization, the whole procedure is fishy right from the start because you go from commuting quantity to anti-commuting quantities, so in which initial order do you replace your symbols ? It is at best ambiguous. In general people choose normal ordering, in which all annihilation operators appears to the right of creation operators, but it could be otherwise and then you would arrive at a slightly different Hamiltonian or Lagrangian. This is related to the "Correspondence Problem", but I don't have the mathematical background to really dig meaningfully into this.
Do you know of any literature on the correspondence problem. The only correspondence problem Im aware of is for computer vision systems. Just seems odd that they are canceling out infinities, mathematically that is. Interesting subject though, and really not as abstract as I thought it would be. The math really isnt so bad so far
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