Second quantization or canonical quantization in quantum field theory and many-body systems is the collective organizing and accounting of an infinity of quantum excitations and their interactions through quantum field operators.

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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 ...
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What is the physical interpretation of second quantization?

One way that second quantization is motivated in an introductory text (QFT, Schwartz) is: The general solution to a Lorentz-invariant field equation is an integral over plane waves (Fourier ...
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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 ...
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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 $x^\...
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Kubo Formula for Quantum Hall Effect

I'm trying to understand the Kubo Formula for the electrical conductivity in the context of the Quantum Hall Effect. My problem is that several papers, for instance the famous TKNN (1982) paper, or ...
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5answers
821 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 ...
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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 ...
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219 views

What do the wave functions associated to the Fock states of each mode of a bound state system mean?

$\renewcommand{\ket}[1]{\left \lvert #1 \right \rangle}$ Consider a string of length $L$ under tension and clamped on each end. This system is described by the wave equation and has a set of modes. ...
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Some questions about anyons?

(1) As we know, we have theories of second quantization for both bosons and fermions. That is, let $W_N$ be the $N$ identical particle Hilbert space of bosons or fermions, then the "many particle" ...
8
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374 views

The Origins of the Second Quantization

I've been studying quantum theory for a while now and have a number of closely related questions that are not giving me any peace. I am not sure if such a long format is appropriate here, but I'd like ...
7
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1answer
152 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
7
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Is it possible to make statements about bosonic/fermionic systems by taking the limit $\theta\to \pi$ or $\theta\to 0$, of an anyonic system?

One might naïvely write the (anti-)commutation relations for bosonic/fermionic ladder operators as limits $$ \delta_{k,\ell} = \bigl[ \hat{b}_{k}, \hat{b}_{\ell}^\dagger \bigr] = \...
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How to understand the entanglement in a lattice fermion system?

Topological insulator is a fermion system with only short-ranged entanglement, what does the entanglement mean here? For example, the Hilbert space $V_s$ of a lattice $N$ spin-1/2 system is $V_s=V_1\...
3
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2answers
613 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 ...
3
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4answers
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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} \begin{...
3
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0answers
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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
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2answers
66 views

Fetter & Walecka's derivation of second quantised potential term in many-particle TDSE

For the potential term in the Hamiltonian, I understand that we go through the same process as with the kinetic energy term. On the RHS of the TDSE, we get something like $\frac{1}{2}\sum_{i}\sum_{j\...
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1answer
133 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'}\\ [...
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Time ordering and Fermions

Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should $T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger]$ return $T[...
0
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1answer
137 views

Perspectives of QFT [closed]

From the answer to this question Computing $\langle0|T[Q(t_2)Q(t_1)]|0\rangle$, I have discovered that there is two perspectives to QFT. I am doing a course which is unfortunately a summary of QFT and ...
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Fetter & Walecka's derivation of second quantised canonical Schrodinger equation for fermions

On page 18, before the occupation number variables for states i and j are changed $n_i \rightarrow n'_i = n_i - 1$ and $n_j \rightarrow n'_j = n_j + 1$ respectively, could we not have rewritten eq. 1....