All Questions
33
questions
0
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1
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67
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Is integral of energy-momentum tensor in QFT over a region $R$ self-adjoint?
Consider a quantum field theory in flat 1+1D spacetime for simplicity. Let $T_{\mu\nu}$ be the conserved symmetric stress tensor. One writes operators by integrating the tensor over the whole space, ...
- 184
5
votes
1
answer
234
views
How does one rigorously define two-point functions?
Let $\mathscr{H}$ be a complex Hilbert space, and $\mathcal{F}^{\pm}(\mathscr{H})$ be its associated bosonic (+) and fermionic (-) Fock spaces. Given $f \in \mathscr{H}$, we can define rigorously the ...
- 1,131
7
votes
1
answer
1k
views
What is the Hilbert dimension of a Fock space?
Quantum field theory in curved spacetimes is often described in the algebraic approach, which consists of describing observables as elements of a certain $*$-algebra. To recover the notion of a ...
- 22.1k
1
vote
0
answers
73
views
Convergence of series of elements in a quasi-local algebra
I am studying the quasi-local algebra on Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics, but there is one thing that is not clear to me at the moment. Let's say that the ...
- 159
3
votes
0
answers
115
views
What is the single-particle Hilbert space in the Fock space of QFT?
In Quantum field theory, the fields are operator-valued functions of spacetime. So for a scalar (spin $0$) field $$\psi: \mathbb{R}^{3,1} \rightarrow O(F),$$ where $O(F)$ is the space of operators on ...
- 51
7
votes
1
answer
321
views
Hilbert space of free theory vs interacting theory
In view of Haag's Theorem, it seems the Hilbert spaces of a free theory and an interacting theory are not the same. Though it seems very believable, I could not find a result that states that this is ...
- 3,350
3
votes
1
answer
223
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What makes coming up with a mathematically solid, non-shaky relativistic quantum field theory (RQFT) so hard?
This is something I know of but I'm not quite sure I understand the details. Particularly, when it comes to interacting RQFTs, such as even QED, where some posts here have pointed out that it cannot ...
- 20.6k
1
vote
0
answers
67
views
Single particle space for the free Euclidean field
In Quantum Field Theory, the free field of mass $m$ can be constructed from creation and annihilation operators on the Fock space. Let $\mathscr{H}_1$ be the single-particle Hilbert space, $F(\mathscr{...
- 121
12
votes
1
answer
982
views
What does mathematical consistency in QFT mean?
My question is more naive than Is QFT mathematically self-consistent?
Just when people talk about the mathematical consistency of QFT, what does consistency mean? Do people want to fit QFT into ZFC? ...
- 141
0
votes
1
answer
116
views
Is this definition of the Fourier Transform of a quantum field operator rigorous?
Let there be a a quantum field operator $\hat\phi(t,\vec{x})$ which, because it acts (pointwise) on a separable Hilbert space, I expect I can write as
$$\hat\phi(t,\vec{x}) = \sum_n\sum_m\phi^n_m(t,\...
- 34
6
votes
2
answers
251
views
How do *-Algebras correspond to operators on a Hilbert space?
In algebraic quantum field theory, a theory is defined through a net of observables $\mathcal{O} \mapsto \mathcal{A}(\mathcal{O})$ fulfilling the Haag-Kastler axioms (see e.g. this introduction, sec. ...
- 1,618
7
votes
1
answer
635
views
What are some good references for field theory via functional analysis?
Many of the aspects of QFT are traditionally done in ways incompatible with a rigorous mathematical treatment, calling for a variety of tricks to fix essentially what was caused by unjustified ...
7
votes
0
answers
1k
views
What is density matrix in QFT?
In quantum mechanics exist fundamental object Density matrix. (See for introduction last chapter in Principles of Quantum Mechanics by David Skinner). Density matrix nesesary to describe systems
even ...
- 5,707
2
votes
0
answers
187
views
How to justify the path integral derivation of the ground state projector?
Usually one argues that the euclidean path integral is able to recover the ground state of a system along the following lines:
Take the time evolution operator $U(t,t_0)=e^{-iH(t-t_0)}$. Transform to ...
- 36.4k
1
vote
1
answer
192
views
In Algebraic QFT, is the state observer dependent?
In the usual approach to QFT presented e.g., in Weinberg's book, the state of a system is dependent on the observer. Quoting this book, in page 109 we have:
Notice how this definition is framed. To ...
- 36.4k