What are the basic mathematical objects in QFT?

[snoopies622]

TL;DR

How are phi and psi (solutions to the Klein-Gordon and Dirac equations) expressed mathematically in quantum field theory?

I found a copy of David McMahon's "Quantum Field Theory Demystified" and I'm already confused on page 4 where he says, " . . in order to be truly compatible with special relativity, we need to discard the notion that $\phi$ and $\psi$ in the Klein-Gordon and Dirac equations respectively describe single particle states. In their place, we propose the following new ideas:
— The wave functions $\phi$ and $\psi$ are not wave functions at all, instead they are fields.
— The fields are operators that can create new particles and destroy particles."

As i understand things,
— the $\psi$ in the Schrödinger equation represents a complex number at every point in space and time, while in the Dirac equation represents four complex numbers at every point in space and time. (I don't know what the $\phi$ in the Klein-Gordon equation represents, but I'm guessing something similar.)
— an operator is something that changes a function into a different function. One way to think about it is - if a function is a vertical list of n complex numbers, then an operator is an nxn matrix that can be multiplied by the column of n numbers to produce a different column of n numbers.

In quantum field theory, what exists at every point in space and time? A matrix? More than one matrix?

 

[malawi_glenn]

I found a copy of David McMahon's "Quantum Field Theory Demystified"

Get a real QFT book.

The "basic objects" in QFT are operator-valued distributions

 

[vanhees71]

My newest discovery is

https://www.amazon.com/dp/9814635502/?tag=pfamazon01-20

It's a gem! It's the best book, which makes relativistic QFT "as simple as possible but not simpler".

 

[malawi_glenn]

My newest discovery is

https://www.amazon.com/dp/9814635502/?tag=pfamazon01-20

It's a gem! It's the best book, which makes relativistic QFT "as simple as possible but not simpler".

Oh you finally got it? Gonna try order it my self :)

 

[dextercioby]

For the people unable to purchase that book, here is a light version of its first half, for free. https://arxiv.org/abs/1110.5013v5

 

[strangerep]

My newest discovery is

https://www.amazon.com/dp/9814635502/?tag=pfamazon01-20

It's a gem! It's the best book, which makes relativistic QFT "as simple as possible but not simpler".

Hmm. Although I've only just now skimmed the first lecture, I will say that I like his style.

 

[Vanadium 50]

Sidney was a genius, and there's a reason why his students populate the theoretical physics departments of so many universities. But I don't think this is the place to start for someone who is just starting out, especially with gaps.

 

[HomogenousCow]

In one particle quantum mechanics we have a system described by a state space and a number of observables like $\hat{q}$, $\hat{p}$, $\hat{S_z}$ etc. At any point in time the system is in some state $|\psi\rangle$ and the wavefunction is given by $\langle q|\psi\rangle$ where $|q\rangle$ are the position eigenstates of $\hat{q}$.

In quantum field theory our system is again described by a state space however now there are observables $\hat{\phi}(\mathbf{x})$, $\hat{\pi}(\mathbf{x})$ for every point in space $\mathbf{x}$. These field observables admit eigenstates $|f(\mathbf{x})\rangle$ for each c-number function of spacetime $f(\mathbf{x})$, so evidently $\langle f(\mathbf{x}) | \psi \rangle$ is not a function but a functional i.e. a function of functions.