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What are the postulates of QFT?by snoopies622
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#1
Mar612, 08:11 PM

P: 611

I like the way quantum mechanics can be expressed as a set of five or six axioms, like in Daniel T. Gillespie's A Quantum Mechanics Primer or David McMahon's Quantum Mechanics Demystified.
Is there a similar set of axioms for quantum field theory? 


#2
Mar712, 09:13 AM

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An example would be Wightman's axioms which are pretty wellknown. See the books by Bogolubov et.al. 1975 or Lopuszanski. There's also an axiomatic formulation due to Haag in terms of (nets of) operator algebras, but this harder to grasp, unless you're also reformulating normal quantum mechanics (the one mentioned in the OP) through operator algebras and their representations.



#3
Mar812, 07:40 AM

P: 611

Thank you.



#4
Mar812, 10:40 AM

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What are the postulates of QFT?
The sad truth therefore is that we currently have no adequate system of axioms for QFT, only a number of ideas how it could possibly look like. Closest to the standard model are in fact not the nonperturbative Wightman axioms but the perturbative EpsteinGlaser approach to quantum field theory; see http://en.wikipedia.org/wiki/Causal_perturbation_theory (and much more by querying scholar.google.com). 


#5
Mar812, 11:42 AM

P: 611

How strange!



#6
Mar812, 01:33 PM

P: 744

When we talk about the axioms of QM, I think of the basic axioms (states in Hilbert space, observables as selfadjoint operators, von Neumann measurements, Schrödinger equation) and not the specific axioms of nonrelativistic single particle QM.
Do the basic axioms really need to be modified for QFT? I thought the problems would occur in in constructing the operators and calculating transition amplitudes. 


#7
Mar912, 09:41 AM

P: 611

I understand the idea of a physical state being represented by a vector and an observable by an operator and how all that works, but if the field is an operator, upon what does it operate? What kind of mathematical object, and what does this object represent? This seems like a completely different formulation to me. 


#8
Mar912, 12:04 PM

P: 744

Let's look at the electron field for example (this is probably flawed but will show the basic features). A simple physical state is a number state n>, where n is the number of electrons you have. The operators of the electron field are ψ(x) and ψ^{+}(x). The first one destroys an electron at space time point x, the second one creates one. The fields themself are not observables, but auxiliary operators from which the observables (like the Hamiltonian) can be constructed. So in general, the field equations (like the Dirac equation) are neither Schrödinger equations (dynamics of the states) nor Heisenberg equations (dynamics of the observables), but dynamical equations for these auxiliary quantities which have no direct physical significance. 


#9
Mar912, 12:04 PM

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#10
Mar912, 07:18 PM

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#11
Mar1112, 03:22 AM

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http://physicsforums.com/showthread.php?t=468492 and threadss citedthere. 


#12
Mar1312, 11:35 AM

P: 611

Thank you kith and A. Neumaier, I think I understand a tiny bit now.



#13
Mar1312, 12:33 PM

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#14
Mar1312, 12:37 PM

P: 611

I'm sorry, I meant  what functions are you referring to?



#15
Mar1412, 06:44 AM

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There isn't any more to the informal notion of ''nice''. 


#16
Mar1412, 01:26 PM

P: 611




#17
Apr312, 06:57 PM

P: 611




#18
Apr1312, 04:32 PM

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