Exploring the Big Picture of QFT

[accdd]

What is the big picture of QFT?

I have studied quantum mechanics from:
-Griffiths
-the first few chapters of Sakurai
-Ballentine
I have studied electrodynamics from Griffiths and General Relativity from Carroll

I have assigned level I to the question, but any answer is welcome

 

[haushofer]

What exactly do you mean by "big picture"?

As I would phrase the conceptual basis of QFT compared to QM: in QM you assign "particles" a wavelike character via de Broglie's relation and the Schrodinger equation, in which the wavefunction tells you the probability to find a particle in a certain state. in QFT you do the opposite: you assign classical waves a particle character by a procedure called "second quantization". The underlying fields are then promoted to "quantum" fields, and the second quantization tells you that "particles" are excited modes of this quantum field. The quantum fields are then operators which act on a vacuum to pop up particles, or on an n-particle state such that some particles disappear. This enables you to describe processes in which particle number is violated but energy is conserved, as Einstein's E=mc2 dictates.

 

[gentzen]

What exactly do you mean by "big picture"? (same as haushofer above)

I am an applied mathematician, so my big picture talks about the math: In non-QFT quantum physics, time is a parameter. In QFT, both time and space are parameters. In the non-QFT Schrödinger picture, only the Hamiltonian operator is time dependent (i.e. dependent on the time parameter). In QFT, many more operators (creation, annihilation, "measurement," ... operators) are dependent on the time and space parameters. But maybe not even space parameters, frequency parameters, or ... make sense too. So we have many different families of operators, which depend on parameters somehow analogous to the time parameter in non-QFT QM.

 

[accdd]

By big picture I mean:
-the things that are there
-the relationships between things
-the fundamental mathematical topics (e.g. what group theory is for in QFT)
-what are the typical applications and problems
-how you would describe QFT in a nutshell
Your interpretations of "big picture" are also fine.

I realized that QFT is a vast and complex field, and unlike NRQM and relativity there is no one way of "understanding things" and that each book has its own approach. All of this confuses me. Why?

 

[WernerQH]

For me, QFT is a machinery for calculating correlation functions. You write down a Lagrangian, integrate over the fields to find the path integral (or some reasonable approximation ), and then obtain the physically interesting quantities by functional differentiation.

Traditionally the starting point is field operators and their equations of motion. But it is just as easy to start from Gaussian functionals (free fields). The confusing thing about field operators is that their equations create the impression of a local and deterministic theory, whereas QFT is surely not a deterministic, but a statistical theory. The field operators represent ensembles of all possible field configurations. While any particular system behaves stochastically, for the averages you can derive deterministic equations.

the things that are there

Very good question! To my mind QFT is not really about field operators. What it is about are events and how they are distributed in spacetime.

My favourite book is "Quantum Field Theory for the Gifted Amateur" by Lancaster and Blundell.

 

[PeterDonis]

-the things that are there

Quantum fields.

-the relationships between things

Lagrangians, which describe the interactions of quantum fields with themselves and each other.

-the fundamental mathematical topics (e.g. what group theory is for in QFT)

The main use of group theory in QFT is with gauge groups for interactions.

-what are the typical applications and problems

Lots of them. The Standard Model of particle physics is the one that probably gets the most press, but it's far from the only one.

-how you would describe QFT in a nutshell

It's the only way we know of to make quantum mechanics consistent with relativity.

 

[A. Neumaier]

The field operators represent ensembles of all possible field configurations.

Operators represent nowhere in QM or QFT ensembles. The ensembles are always described by the states.

A state of a quantum field specifies the $N$-point functions - the only quantitative output of QFT, from which one gets observable items such as macroscopic fields and collision cross sections - at all spacetime positions.

But there is only a single realization of spacetime that physicists have access to - it is called the universe. This implies that one cannot create an ensemble of realizations of a QFT. Hence the $N$-point functions of QFT must make assertions about what happens in this single realization.

 

[Vanadium 50]

By big picture I mean:
-the things that are there
-the relationships between things
-the fundamental mathematical topics (e.g. what group theory is for in QFT)
-what are the typical applications and problems
-how you would describe QFT in a nutshell
Your interpretations of "big picture" are also fine.

All in a few hundred word post.

I realized that QFT is a vast

Do you?

I suspect that, despite people trying their best, you will not be satisfied,

 

[accdd]

References, links, notes, textbooks, papers, videos, etc. are welcome.

Vanadium 50
Excuse me if my questions are naive.
If this or other threads started by me are inappropriate, feel free to delete them.
If you think I am contributing negatively on this site feel free to delete my account.

 

[PeterDonis]

References, links, notes, textbooks, papers, videos, etc. are welcome.

If you mean general references about QFT, that's fine. Just be aware that your question is extremely open-ended and subjective, so it is not likely to lead to much useful discussion in this thread.

 

[PeterDonis]

Moderator's note: Thread moved to the interpretations subforum, since answers to the "big picture" question will be different for different intepretations.

 

[Vanadium 50]

answers to the "big picture" question will be different for different intepretations.

I don't think so. QFT has largely avoid the interpretation nonsense that has plagued amateurs and QM.

But in any event...

There exist things that snswer the OP's "big picture" questions. We call them "books". They have approximately the same number of pages as posts have words. I don't think the required level of consensation will be acceptable to anyone participating.

Oh, and this is where someone says "If you really understood it, you could explain it to your grandmother [in a forum post]". Don't. Just don't.

 

[topsquark]

...
Oh, and this is where someone says "If you really understood it, you could explain it to your grandmother [in a forum post]". Don't. Just don't.

Didn't Einstein say that about 6 year olds? :)

-Dan

 

[Fra]

It's the only way we know of to make quantum mechanics consistent with relativity.

I second this. From the perspective of evolving a theory i think starting with normal QM, and then concepts such as indistinguishable and the lorentz symmetry constraint almost gives birth to many qft concepts such as antiparticles and one extra level of reflection of probability via 2nd quantization. I think the old way of introducing qft via iterates (2nd) quantization providea more insight than the more polished axiomatic introduction. The latter may be cleanr from the perspective of mathematics... but the axiomatic approach seem hars ro merge with the agent centeree view. I see nth quantization conceptully as a sort of "self reflection" in highee order processing that the agent are forces to in find and information preserving representation.

I also think that the foundational issuea in QM essentiall remains in QFT. So its a challenge of internal consistency to not only inyerpret NRQM but also QFT.

/Fredrik

 

[Demystifier]

What is the big picture of QFT?

This question can be answered from many different points of view.

One possible view, which you may or may not like, is the Bohmian view. The big picture (i.e., without technical details) of QFT from the Bohmian point of view is presented in https://arxiv.org/abs/2205.05986.