I'm seeing lots of underlying connections between the canonical formulism of QFT and QM. But I'm getting a bit confused by their differences. I'll just write down my thought process:(adsbygoogle = window.adsbygoogle || []).push({});

QM is a one parameter system (t) in a space with three quantized operators (x,y,z)

QFT is a four parameter system (t,x,y,z) in a space with one/four/four quantized operators (scalar/spinor/vector).

In QM we develop the hamiltonian formalism and treat our one parameter as "special."

In QFT we develop the hamiltonian formlism and treat one of our four "equal" parameters as special. And I lose sleep at night because of it. Anyways...

In QM we have the S/H pictures where we apply the development of our parameter t to the states/operators.

In QFT we treat our special parameter with privilege once again and apply the development of t to the states/operators.

In QFT, I think, but I'm not sure, that the rest of the Schrodingafication/Heisenbergation is assumed throughout and we apply all the rest of the parameter dependence upon the field operators. IE ∂ψ/∂x^i = 0 for all states ψ and i=1,2,3.

I actually think typing this up has cleared up my thoughts well enough, but I'm still inquiring further as I read my textbook. But I can't find anybody that develops these concepts AT ALL. Maybe an obscure mention, but not completely.

Also, I posted this earlier as I wasn't quite as far along in my logical development. I'll just repost it here if anybody has any desire to respond to it.

I'm a little confused in the allocation of t,x,y,z dependence between the states and fields in all the different pictures.

In QM, we had a state ψ and an operator A. In the SP, ψ is a function of t and A isn't. In the HP, A is a function of t and ψ isn't.

In QFT, we have a state ψ and an operator A. But we also give ourselves three new dependent variables for our system to do whatever it wants with; x,y,z. I don't see a reason why we don't have xyzHeisenberg pictures and xyzSchrodinger pictures. Not necessarily saying they would be useful, but their complete lack of mention in P&S, Zee, Srednicki, Zuber etc have me puzzled. Can't we pick between an operator relation of any of these?

A(0)ψ(t,x,y,z) --- A(x,y,z)ψ(t) --- A(t)ψ(x,y,z) --- A(t,x,y,z)ψ(0)

Also, if anybody has a resource that talks about this, I'd like to read it.

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# Trying to make the connection with QM as a 1 parameter QFT

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