# What are the basic mathematical objects in QFT?

• snoopies622
These functional fields are what we want to describe as our wavefunction. The key point is that the wavefunction is not a function at a single point in space, it is a function at every point in space. The wavefunction is given by a functional field equation in which the coefficients depend on the position and momentum of every particle in the system. In summary, 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 aref

#### snoopies622

TL;DR Summary
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 Schrodinger 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?

Last edited:
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

• aaroman and dextercioby
• strangerep, PeroK, malawi_glenn and 3 others
• vanhees71
Last edited:
• vanhees71, strangerep and topsquark

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.

• vanhees71
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.

• vanhees71 and malawi_glenn
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.

• topsquark, vanhees71, PeroK and 1 other person