1
vote
1answer
76 views

Annihilation Operator on the Fock space

I agree that $$\hat a|0\rangle=0$$ But then, based on the above, the following should hold $$\hat a_k |N_1,...,N_{k-1},0,N_{k+1},...\rangle=|N_1\rangle\oplus\cdots\oplus |N_{k-1}\rangle\oplus \hat ...
1
vote
1answer
68 views

Hermiticity of the quantum field

The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
1
vote
0answers
66 views

Normal ordering

If I understood correctly there are two terms called normal ordering: $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the ...
9
votes
3answers
257 views

In what sense is a quantum field an infinite set of harmonic oscillators?

In what sense is a quantum field an infinite set of harmonic oscillators, one at each space-time point? When is it useful to think of a quantum field this way? The book I'm reading now, QFT by ...
2
votes
1answer
97 views

Schrödinger evolution for a Klein-Gordon equation

I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with ...
1
vote
3answers
224 views

Bogoliubov transformation with a slight twist

Given a Hamiltonian of the form $H=\sum_k \begin{pmatrix}a_k^\dagger & b_k^\dagger \end{pmatrix} \begin{pmatrix}\omega_0 & \Omega f_k \\ \Omega f_k^* & \omega_0\end{pmatrix} ...
3
votes
0answers
198 views

Is non-relativistic quantum field theory equivalent with quantum mechanics?

Related post Can we "trivialize" the equivalence between canonical quantization of fields and second quantization of particles? Some books of many-body physics, e.g. A.L.Fetter and ...
1
vote
1answer
73 views

Deriving commutation relations in second quantisation

I am trying to start from: \begin{align*} [\phi(x),\pi(x')] = i\hbar\delta(x-x') \\ [\phi(x),\phi(x')] = [\pi(x),\pi(x')]=0 \end{align*} to derive: \begin{align*} [a(k),a(k')^\dagger]=\delta_{kk'}\\ ...
2
votes
3answers
261 views

Dirac equation in QFT vs relativistic QM

How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The ...
0
votes
1answer
197 views

Second quantization of Klein Gordon field

Does the second quantization of the Klein Gordon field which involves using the harmonic oscillator paradigm ultimately lead to the conclusion that electromagnetic field is nothing but photons(bosons) ...
2
votes
3answers
206 views

Schroedinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as ...
12
votes
5answers
492 views

Can we “trivialize” the equivalence between canonical quantization of fields and second quantization of particles?

As Weinberg exposited in his QFT Vol1, there are two equivalent ways of arriving at the same quantum field theories: (1). Start with single-particle representations of Poincare group, and then make a ...
2
votes
1answer
74 views

Fock state and corresponding relations for continuous momentum label

In Wikipedia I found following relation for Fock state: $$ \hat {a}_i| \{n_j\}_j\rangle ~=~ \sqrt{n}_i| \{n_j-\delta_{ij}\}_j\rangle, $$ where $n_j$ refers to the number of $j$'th particles. This ...
1
vote
1answer
80 views

Matrix elements of a one-fermion operator (first and second quantizations)

I'm currently struggling with the expression of operators in second quantization. I did an exercise in which I had to consider a fermion in a central potential $V(\vec{r})$ and show that the matrix ...
3
votes
0answers
148 views

Fock Subspaces and Weight Vectors

This is my first time taking a physics course (I'm a mathematics major), so I'm encountering a lot of new things, which I'm kind of expected to know. In particular, how to work with Bosons. I've got ...
2
votes
2answers
139 views

Scalar QFT Fock Space

I want to demostrate the following relation of the normal ordered product: $\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$ I proved the commutation relation ...
2
votes
1answer
80 views

A change of sign in the electron-hole second quantization form

It is common to see people do a change of sign in the so called electron-hole representation, namely, $$ b^{\dagger}_{-k}=a_{v,k} $$ similar argument also seen in 1992 mattuck's book "guide to ...
0
votes
2answers
378 views

Occupation Number Representation in Second Quantization Formalism — What do the entries mean?

I'm reading about the second quantization formalism. I can see the advantages of using number states to represent multiparticle states. Here's my question: Let's say we're given a single-particle ...
2
votes
0answers
93 views

Second quantization with qubits

Is "second quantization" means system wich can contain variable, unknown, superposed and otherwise uncertain number of qubits? Can "second quantized" system contain 0.5% of 1 qubit and 95% of 2 ...
4
votes
2answers
297 views

Second Quantization - Texts

I am trying to familiarize myself with the ideas of Second Quantization. However, the literature that I can find online seems only to outline the tools of this formalism of quantum mechanics. There ...
2
votes
1answer
168 views

Reason behind canonical quantization in QFT?

Reason behind canonical quantization in QFT? In the scalar field theory we simply promote the scalar field, $\phi(x)$ to a set of operators: $\hat{\phi}(x)$. What is the reason behind this?
2
votes
1answer
310 views

Density operator in second quantization

I would want to understand why the density operator in second quantization takes the form: $$\rho_\sigma(\mathbf{r})=\Psi_\sigma^\dagger(\mathbf{r})\Psi_\sigma(\mathbf{r})?$$ Is this a definition or ...
1
vote
3answers
257 views

EM field quantization

I'm trying to quantize the electromagnetic field by solving the vector potential wave equation, that is: $$\nabla^{2} \mathbf{A} = \dfrac{1}{c^{2}} \dfrac{\partial ^{2} \mathbf{A}}{\partial t^{2}}, ...
11
votes
3answers
1k views

What does the ordering of creation/annihilation operators mean?

When a system is expressed in terms of creation and annihilation operators for bosonic/fermionic modes, what exactly is the physical meaning of the order in which the operators act? For example, for ...
8
votes
1answer
605 views

Quantization of Gravitational Field: Quantization conditions

I'm begining to study Quantization of field with the second quantization formalism. I've studied phononic field, electromagnetic field in the vacuum and a generic relativistical scalar field. I ...
8
votes
2answers
793 views

What does second quantization mean in the context of string theory?

String field theory (in which string theory undergoes "second quantization") seems to reside in the backwaters of discussions of string theory. What does second quantization mean in the context of a ...
33
votes
9answers
2k views

Is a “third quantization” possible?

Classical mechanics: $t\mapsto \vec x(t)$, the world is described by particle trajectories $\vec x(t)$ or $x^\mu(\lambda)$, i.e. the Hilbert vector is the particle coordinate function $\vec x$ (or ...