All Questions
Tagged with quantum-field-theory hilbert-space
681
questions
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40
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About momentum states covariant normalization
I'm following QFT of Schwrtz and I have a doubt about Eq. (2.72).
In particular, from Eq. (2.69): $$[a_k,a_p^\dagger]=(2\pi)^3\delta^3(\vec{p}-\vec{k}),\tag{2.69}$$ and Eq. (2.70): $$a_p^\dagger|0\...
- 147
2
votes
0
answers
60
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Asymptotic states and physical states in QFT scattering theory
Context
In the scattering theory of QFT, one may impose the asymptotic conditions on the field:
\begin{align}
\lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
- 108
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0
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31
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Why the Slavnov operator is self-adjoint? [duplicate]
In the context of BRST we can define the Slavnov operator $\Delta_{BRST}$ which generates BRST transformations. My lecture notes claim that $\Delta_{BRST}$ is self-adjoint, but I don't see why.
- 357
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2
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116
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Do different bases of Fock space commute?
$\newcommand\dag\dagger$
Suppose we have a Fock space $\mathcal{F}$ with two different bases of creation and annihilation operators $\{a_\lambda, a^\dag_\lambda\}$ and $\{a_{\tilde \lambda}, a^\dag_{\...
0
votes
1
answer
44
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Bogoliubov transformation of Bunch-Davies vacuum
Let $\left|0\right>$ be the Bunch-Davies vacuum state of a QFT, for example a free scalar field, in de Sitter space. The creation and annihilation operators w.r.t. this state is a vacuum, i.e. $a^...
- 505
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2
answers
58
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How can I construct a trivial product state in the continuum?
When working on the lattice it is easy to define a trivial product state. A state $|\psi\rangle$ is a trivial product state if it admits the following tensor decomposition,
\begin{equation}
|\psi\...
1
vote
0
answers
90
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In the path integral formulation of QFT, how do we get quantized particles out of a field?
Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and ...
- 1,262
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1
answer
79
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Regarding Energy Eigenstate and Position Eigenstate
I am solving problem 14.4. (a) of Schwartz's Quantum Field Theory and the Standard Model. It is related to the simple harmonic oscillator in quantum mechanics. It asks the eigenstate of the position ...
- 79
1
vote
0
answers
74
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Motivation behind reflection positivity
I have taken a look at this physicsSE question, Wikipedia, and this paper by Jaffe which go over reflection positivity. While they all nicely explain the definition behind reflection positivity and ...
- 3,350
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26
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How can interacting field operators in $2D$ still satisfy the canonical commutation relation?
Free fields in any dimensions are well-known to be Gaussian, act on the Fock space and satisfy the canonical commutation relations.
By definition, interacting field operators are NOT such cases, as ...
- 1,669
1
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1
answer
54
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Choice of spacetime foliation while quantising a conformal field theory
I was reading Rychkov's EPFL lectures on $D\geq 3$ CFT (along with these set of TASI lectures) and in chapter 3, he starts discussing radial quantisation and OPE (operator product expansion). I ...
- 904
6
votes
4
answers
623
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How can a QFT field act on particle states in Fock space?
Recently I asked a question that was considered a duplicate. However I felt that the related question didn't answer my doubts. After a bit of pondering I have realized the core of my discomfort with ...
- 4,577
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1
answer
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
4
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0
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106
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How to interpret QFT fields (in relation with QM)? [duplicate]
In QM we deal with the Schrödinger equation:1
$$i\frac{\partial}{\partial t}\psi = H \psi$$
the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
- 4,577
1
vote
1
answer
65
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Parity operator action on quantized Dirac field
I am stuck on equation 3.124 on p.65 in Peskin and Schroeder quantum field theory book.
There they are claiming that:
$$P\psi(x)P=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf p}}}\...
- 413
1
vote
3
answers
154
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What does the state $a_k a_l^\dagger|0\rangle$ represent?
Consider the action of the operator $a_k a_l^\dagger$ on the vacuum state $$|{\rm vac}\rangle\equiv |0,0,\ldots,0\rangle,$$ the action of $a_l^\dagger$ surely creates one particle in the $l$th state. ...
- 11.8k
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0
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39
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Normalization of one particle state wave function in fock space - commutator
In deriving the 1/$\sqrt{N!}$ normalization factor the first step is looking at the one particle state (see image below). I am confused about how we got from the first line to the second? Maybe I am ...
2
votes
1
answer
225
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Poincaré invariance and uniqueness of vacuum state
I'm trying to understand the Poincaré invariance of the vacuum state in Minkowski spacetime, how it implies the uniqueness of the vacuum state, and why there's no unique vacuum state in general ...
5
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1
answer
234
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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
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0
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75
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What does a quantized field in QFT do? [duplicate]
I'm studying for an exam called Introduction to QFT. One of the main topics in this class is the quantized free fields.
I can now find the fields that solve the Klein-Gordon equation and the Dirac ...
- 19
3
votes
0
answers
64
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Operator that gives a permutational symmetry factor
Suppose that we have a system with $N$ bosonic modes, meaning that there is a vacuum state $|0\rangle$ and a set of $N$ pairs of creation-annihilation operators $a_i$ and $a^{\dagger}_i$. When ...
- 31
-3
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2
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107
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Multi-particle Hamiltonian for the free Klein-Gordon field
The text I am reading (Peskin and Schroeder) gives the Hamiltonian for the free Klein-Gordon field as:
$$H=\int {d^3 p\over (2\pi)^3}\; E_p\; a^{\dagger}_{\vec p}a_{\vec p}$$
This does not seem to be ...
- 4,399
-2
votes
1
answer
74
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On creation annihilation operators of the free Klein-Gordon field [closed]
I want to calculate multiparticle states like $|\vec p,\vec p\rangle$ from $|0\rangle$. It seems that I would need to compute from things like: $a^{\dagger}_{\vec p}a^{\dagger}_{\vec p}|0\rangle$?
It ...
- 4,399
3
votes
0
answers
64
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Deriving a contradiction from the LSZ condition
I'm reading the LSZ reduction formula in the wikipedia:
https://en.wikipedia.org/wiki/LSZ_reduction_formula
To make the argument simple, let $$\mathcal{L}=\frac{1}{2}(\partial \varphi)^2 - \frac{1}{2}...
- 61
3
votes
2
answers
148
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Algebraic QFT from a Lagrangian
In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
- 368
7
votes
1
answer
1k
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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
2
votes
1
answer
116
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What's the exact definition of fields in conformal field theory?
For example we work with a 2d scalar field $\phi$. I guess $\phi$, $\partial_z\phi$, $\partial_{\bar z}\phi$ are fields, are there more? Is it true that all fields are in the form of $\partial_z^i\...
- 249
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1
answer
85
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What determines the conjugation of a state in quantum field theory?
In basic quantum mechanics, we define the inner product between two states $\phi$ and $\psi$ as $\phi^\dagger \psi$, where $\phi^\dagger$ is the conjugate transpose of the vector $\phi$. However in ...
- 2,504
4
votes
1
answer
91
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State-Operator Correspondence and symmetry in CFT in general dimension
Let us assume to have a QFT ($\mathcal{L}$) with translational, Lorentz, scale and conformal invariance.
I ask because we can, for example the free scalar free theory, canonically quantize the system ...
- 194
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votes
1
answer
154
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Peskin & Schroeder equation (7.2)
I found this completeness relation of momentum eigenstate $|\lambda_p\rangle$
Here $|\Omega\rangle$ is the vacuum, and $|\lambda_p\rangle$ represents the state with one particle labeled by $\lambda$ ...
- 1
2
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0
answers
77
views
LSZ theorem for trivial scattering
The $1\to1$ scattering amplitude is trivial and is given by (take massless scalars for simplicity)
$$
\tag{1}
\langle O(\vec{p}) O^\dagger(\vec{p}\,')\rangle = (2 | \vec{p}\,|) (2\pi)^{D-1} \delta^{(...
- 181
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0
answers
81
views
Why Fock representation holds only in a free quantum field theory?
With a quantum system with $N$ degrees of freedom, all the representations are unitarily equivalent to Fock representation. However, if the number of degrees of freedom goes to infinity, there are ...
- 159
-3
votes
1
answer
91
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Some calculation in Mahan book, p73 [closed]
On page 73 of Mahan, Many-particle physics, 3rd edition, one finds
$$
_0\langle|S(-\infty,0) = e^{-iL}_0\langle|S(\infty,-\infty)S(-\infty,0).
$$
I'm wondering why this is true, as in the previous ...
- 793
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1
answer
117
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Problem with understanding the concept of vacuum state of a quantum field
The vacuum state is the state with the minimum energy, which implies no excitations, which I assume is the same as a state with no particles. Then I am confused about a static electric coulomb field. ...
- 795
9
votes
4
answers
3k
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If quantum fields are operator valued distributions, why aren't they always smeared?
I don't completely understand the distributional character of a quantum field because I never see them "smeared" in basic textbooks. As I understand it, if $\chi : \mathcal{F} \rightarrow \...
- 615
-1
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1
answer
249
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What does the field operator $φ(x)$ do to the Fock space?
For simplicity: imagine a free, scalar theory, and a 1 particle universe.
Spacetime: we have an operator $φ(x)$ defined everywhere on spacetime.
Fock space: the space of states in which the particle ...
- 454
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votes
1
answer
114
views
Can we construct the QFT Fock space with only field operators $φ(x)$ acting on the vacuum?
We always hear that
The Fock space is constructed with multiple $~a^\dagger_{\vec p}$ acting on the vacuum for different values of ${\vec p}$ (we can use alternatives notations to ${\vec p}$ because ...
- 454
-2
votes
1
answer
144
views
Does the QFT Klein-Gordon equation describe the state of the field or the field operator?
In the canonical quantization of QFT we talk about:
states representing a field.
field operators.
The quantum Klein-Gordon equation is expressed in terms of the field φ. Is φ (in the equation) the ...
- 454
3
votes
1
answer
153
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Equivalence of Schrödinger operator formalism and Path Integral Formulations for Scalar Field Theory
I'm exploring the deep connections between different formulations of quantum field theories and have a specific question about the equivalence between the Schrödinger representation and the path ...
- 1,548
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votes
1
answer
64
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The relation between spin and momentum in particle state
In quantum field theory, it seems that when we consider a massive particle's spin degree of freedom, we usually do in the particle's rest frame. And I know the little group will only change spin DOF ...
- 35
2
votes
1
answer
284
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How to derive the Fermion generating function formally from operator formalism?
The generating functionals for fermions is:
$$Z[\bar{\eta},\eta]=\int\mathcal{D}[\bar{\psi}(x)]\mathcal{D}[\psi(x)]e^{i\int d^4x
[\bar{\psi}(i\not \partial -m+i\varepsilon)\psi+\bar{\eta}\psi+\bar{\...
- 619
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0
answers
73
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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
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votes
1
answer
106
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How does Weinberg definition of particle states from standard momentum work?
In his first volume, part 2.5, Weinberg define one particle states $Φ_{p,𝜎}$ ($p$ is the momentum and $𝜎$ another quantum number) in the following way :
Choose a Standard momentum $k$
Find a ...
- 43
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0
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93
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How do Poincare group act on Classical field, Quantum field operator, Field configuration states, Fock space states?
I will try to make each of my statement as clear as possible, if any of the statements are wrong prior to my question, please point them out:)
For simplicity, we work in free QFT with scalar field.
...
- 21
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1
answer
176
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Dimensionality of state space of TQFTs
As the title suggests, I am wondering about the dimensionality of state spaces in $d$-dimensional TQFTs. As of yet I have mostly been concerned with the mathematical, functorial definition of TQFTs as ...
0
votes
1
answer
59
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Concrete understanding of QFT Hilbert space for spinor
I'm trying to understand the concepts of a spinor field in QFT. I naively understand there are two values at each spatial position $\vec{r}$: a probability amplitude and a spinor value. Is there a ...
- 148