All Questions
Tagged with quantum-field-theory hilbert-space
681
questions
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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
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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_{\...
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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
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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
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0
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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
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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
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0
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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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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
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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
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votes
4
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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
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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
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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$$
the wave function $\psi(x)$ is the main object of interest: it can be interpreted as a scalar field, in the ...
- 4,577
1
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1
answer
65
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Parity operator action on quantized Dirac field
I am stuck on equation 3.124 on p.65 in Peskin and Schroeder quantum field theory book.
There they are claiming that:
$$P\psi(x)P=\displaystyle\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf p}}}\...
- 413
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3
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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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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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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
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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 ...
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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
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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
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0
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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
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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
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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
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1
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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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1
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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