What are the basic mathematical objects in QFT?

• snoopies622
In summary: 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 are
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?

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snoopies622 said:
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
vanhees71 said:

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

1. What is QFT?

QFT, or quantum field theory, is a theoretical framework used to describe the behavior of particles at the subatomic level. It combines elements of quantum mechanics and special relativity to explain the interactions between particles and their corresponding fields.

2. What are mathematical objects in QFT?

The basic mathematical objects in QFT are fields, operators, and states. Fields describe the properties of particles, operators represent physical observables, and states represent the possible configurations of a system.

3. How are fields represented in QFT?

In QFT, fields are represented by mathematical functions that describe the values of a particle's properties at different points in space and time. These functions are called field operators.

4. What is the significance of operators in QFT?

Operators in QFT are used to calculate the values of physical observables, such as energy or momentum, for a given state of a system. They also play a crucial role in the mathematical formulation of quantum mechanics and are essential for understanding the behavior of particles at the subatomic level.

5. How do states relate to mathematical objects in QFT?

States in QFT represent the possible configurations of a system, including the arrangement of particles and their corresponding fields. They are described by mathematical objects called wave functions, which are used to calculate the probability of a particle having a particular property at a given time and location.

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