@Ujjal
The basic idea of QFT is to solve an initial-value problem for an infinite set of operators. We thus assume that we know the field operatos (as we call the operators under consideration) an one instant of time (e.g. for t=0, or, more elegantly, we specify the field operators on a 'space-like hyperplane'). What does it mean to specify a infinite set of linear operators in a Hilbert space? I is a much more demanding task than, for instance, to specify the initial condition for a sound wave (where a scalar pressure field and two vector fields, for velocity and time derivative of velocity is all we need). As the history of quantum theory teaches, an efficient way to specify systems of operators is to specify their commutation relations (think position and momentum operators, the components of angular momentum,...). In the best case, we specified commutation relations in a way that there is, up to unitary equivalence, a single system of operators which satisfies our specification. The next step then is to define a Hamiltonian H as a function of the field operators (all at t=0 !). Defining operators for times t>0 by A(t)= exp(-itH)A(0)exp(itH) makes a dynamical system - our QFT. Since H is a function of all field operators for t=0, it is clear that the commutator [A(t),A(0)] is an extremely complicated function of t=0 field operators. It is only accessible through approximations or through computer programs. The problem with this approach is that it always results in non-convergent infinite sums if we work in a space which has infinitely many discernible points. Since discrete finite space approximates the continuum to any required accuracy it is at least an instructive thought experiment to approach QFT that way (as done in the classical text books like Schiff and Bjorken/Drell).
I hope it is not an aside to consider the meaning on such nonequal times results. The need for many times in quantum theory has been discussed before. The spatial coordinates must be manifold since correlations exist and relativity suggests we must have many time labels as well. The meaning of this has never been clear and we seem to have not measurement procedure to pull out much meaning from them. If we use a Schrodinger-type approach to field theory, then the reality of many times is more evident. The only solution I ever found was to consider the subset of wavefunctions in Fock space that can correspond to classical objects which then naturally induce an analog of a spacelike hypersurface with a single cone of timelike vectors at each point.
Interesting question. If you suppress the spatial coords and create a pure spin system you can describe any spin system with products of SU(2). This was one of Schwinger's results. Does this get at what you are asking?
I think I may have misinterpreted your question. The many times feature can be computed for such a separable spin system but it is not that interesting. Embedded in QFT is the use of many time system during the freely evolving paths between interactions but it is never explicit. I don't see why SU(2) has any special role for this.
To address this question one has to clearly see in which picture one is working (i) Schrodinger picture or (ii) Heisenberg picture. In the Schrodinger picture operators are time independent whereas state vectors contain time dependence whereas in the Heisenberg picture operators are time dependent whereas the state vectors are not.
The canonical quantization prescription elevates observables to operators and postulates commutation relations among them. In the Schrodinger picture the postulates are,
[φa(x), φb(y)] = [πa(x), πb(y)] = 0
[φa(x), πb(y)] = i δ3(x − y) δab
In the Heisenberg picture, the two operators defining commutation relations depend on time, say t1 and t2. The point is that these commutation relations are valid in the Heisenberg picture only when t1=t2. Therefore they are equal time commutation relations. This relation is quite useful in the Heisenberg picture because we know how operators evolve with time. They follow the Heisenberg's equation of motion,
dA(t) / dt= i/hbar [H, A(t)]
In addition there is a (del A/del t) term for operators having explicit time dependence.
Probably it puts things into perspective to recall the old Lehman-Källe´n spectral representation of the vaccum expectation value of the commutator of two field operators at arbitrary space-time positions (thus not only equal time). The Schweber QFT-textbook has a long chapter on it (omitting Danish Källe´n).
Schweber is a great reference for such things. Still my favorite QFT book though Weinberg is pretty extensive.
Clifford, your wording confuses me, a non-native speaker of American. Do your words say that you prefer Schweber (over Weinberg) ?
But we need not use equal time commutation relation! Instead, we can start, I believe, with the commutators [Φ(x), Φ(y)]= i∆(x,y), where x and t are arbitrary events in the 4D world
I think, both the commutator and the propagator are auxiliary quantities without any particular physical meaning
@Ulrich, I liked to structure of Schweber's book since it emphasized real space features and is much more old school. Weinberg is a very different one (well three!). QFT was always more formulaic than conceptual but Feynman didn't want it that way. Dyson and others pushed it that direction and by the 70's people weren't really trying to do much more than get calculations done with slicker mathematical tricks so the later work reads more like math than physics. That is all I meant. There certainly was progress but the older stuff was always more inspiring to me.
@Niltopal, Here is a link to Schwinger's famous paper. http://www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf. I kind of hate it but I don't claim anything in it is actually wrong. I recently made some progress on many body times in regards to QFT and eliminating the need for operators and commutation relations. It is not short enough and also not in depth enough, in my opinion, so has some room for improvement but has the basic ideas.
Article Beyond Quantum Fields: An Operator-Free Covering Theory for QED
@ Nilotpal,
Commutation relations between fields may involve i. Observable quantities, however, are represented by Hermitian operators which satisfy, A†=A
Operators are again abstract quantities which act on an abstract space of eigenfunctions. However if we want to calculate some useful quantities we have to consider representations of operators.
The most common representation in quantum field theory uses actions of creation and destruction operators. A creation operator adds one quantum in a system and destruction operators reduces one. If you start from the vacuum state, denoted by |0> and start building your eigenstates by using creation operators, you will end up with many different possible states in which the system can exist. These states can be labeled by the eigenvalues of the number operator. Collection of all these possible states of a system forms a complete basis which is used for operator representation and calculation
These issues are explained in detail in the book by Peskin and Schroeder.
@Serguei,
what you write down is valid for the free scalar field. Any attempt to include interaction will spoil the validity of your nice formula.
Ulrich,
Yes, it probably will (beauty is so fragile!) Then again, is there something it will not spoil (beyond "the physical level of rigour")?
In QFT, one Usually, starts with equal time commutation relations and the can derive the commutator between operators. On the other hand, one can start with commutators and derive the commutation relations. Its like the Egg and the chicken ( which comes first).
According to Heisenberg uncertainty principle we can not measure position and momentum of a particle at the same time accurately. This principle is described by commutation relation of canonical position and momentum of quantum fields. Thus this commutation relation must be of equal time, because we can not measure position and momentum at same time accurately. But we can measure position and momentum accurately at different times, that what Heisenberg principal says. For example we can measure position at one instance of time and then measure momentum after some instance of time accurately. Thus the value of commutation relation of position and momentum at different times will be zero.
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