Questions tagged [second-quantization]

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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47
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9answers
6k 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 $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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3answers
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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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3answers
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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 ...
18
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0answers
271 views

Can a theory gain symmetries through quantum corrections?

It is well known that not all symmetries are preserved when quantising a theory, as evinced by renormalising composite operators or by other means, which show that quantum corrections may alter a ...
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5answers
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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 ...
16
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3answers
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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 ...
16
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3answers
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Correct way to write the eigenvector of a diagonalized hamiltonian in second quantization

I am studying diagonalization of a quadratic bosonic Hamiltonian of the type: $$ H = \displaystyle\sum_{<i,j>} A_{ij} a_i^\dagger a_j + \frac{1}{2}\displaystyle\sum_{<i,j>} [B_{ij} a_i^\...
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1answer
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Horrifying electron gas model

I am given the Hamiltonian, in an exercise called plasmons, and where $\langle, \rangle $ denotes the expectation value. $$ H = \sum_{k} \varepsilon_k a_k^{\dagger} a_k + \sum_{k_1,k_2,q} V_q a_{k_1+...
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Bogoliubov transformation is not unitary transformation, correct?

To diagonalize quadratic term in the antiferromagnet Heisenberg model, we may introduce the Bogoliubov transformation:$a_k=u_k\alpha_k+v_k\beta_k^\dagger$, $b_k^\dagger=v_k\alpha_k+u_k\beta_k^\...
14
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1answer
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Schrödinger wavefunctional quantum-field eigenstates

The reason that I have this problem is that I'm trying to solve problem 14.4 of Schwartz's QFT book, which've confused me for a long time. The problem is to construct the eigenstates of a quantum ...
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2answers
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Why do we need to embed particles into fields?

In QFT we have the so-called embeding of particles into fields. This is discussed at full generality in Weinberg's book, chapter 5. In summary what one does is: From Wigner's classification, for each ...
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Why do we need $2^\text{nd}$ quantization of the Dirac equation

As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion about it. So let me start with its definition: The Dirac equation is given by, $$ i \hbar ...
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5answers
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Quantizing a complex Klein-Gordon Field: Why are there two types of excitations?

In most references I've seen (see, for example, Peskin and Schroeder problem 2.2, or section 2.5 here), one constructs the field operator $\hat{\phi}$ for the complex Klein-Gordon field as follows: ...
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2answers
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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 ...
12
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1answer
290 views

Quantum Fields as Functionals

In single-particle quantum mechanics, particles are replaced by wave-functions -- which are functions from the space of possible particle positions to the complex numbers. It seems that the most '...
11
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3answers
754 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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2answers
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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 ...
10
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335 views

Finding the spectrum of a polynomial of the creation and annihilation operators

Is there a general algorithm to find the spectrum of $S S^\dagger$, where $S$ is a homogenous polynomial (of degree $n$) of the annihilation operators (of $d$ variables)?
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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" ...
10
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1answer
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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 J.D....
9
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430 views

Scalar product between Fock states

Suppose to have a chain (of size $L$) with bosons, and $\hat{a}_i^\dagger$,$\hat{a}_i$ are the associated creation and annihilation operators at site $i$. A Fock state can be written as: \begin{...
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Book recommendations for second quantization

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. ...
9
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2-nd quantized TQFT formalism?

Suppose that we have a certain TQFT in the Atiyah-Singer sense. It is given by a functor $Z$ which associates: To connected oriented $n-1$-manifolds $a, b, \dots$ (in what follows called compact ...
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1answer
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Particle density operator in second quantization form

The particle-density operator is given by $n(\mathbf{x})=\sum_{\alpha}\delta^{(3)}(\mathbf{x}-\mathbf{x}_{\alpha})$, then how to derive its representation in terms of creation and annihilation ...
8
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1answer
410 views

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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2answers
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QM Continuity Equation: Many-Body Version for Density Operator?

I am trying to brush up my rusty intuition on second quantization and many-particle systems and i came across the following problem: In 1-particle QM we have the continuity equation $$ \frac{\...
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3answers
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How exactly is “normal-ordering an operator” defined?

(In this question, I'm only talking about the second-quantization version of normal ordering, not the CFT version.) Most sources (e.g. Wikipedia) very quickly define normal-ordering as "reordering ...
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1answer
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Second Quantization in Condensed Matter and Quantum Field Theory

There appears to be an apparent dichotomy between the interpretation of second quantized operators in condensed matter and quantum field theory proper. For example, if we look at Peskin and Schroeder, ...
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3answers
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If all particles are fields, why does first quantization work for some particles?

After a lot of Google and asking professors about the two quantization methods, I have learned that first quantization is what you use to quantize classical particles, while second quantization is ...
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1answer
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First quantization vs second quantization

What is the difference between first quantization and second quantization and where does the name second quantization come from?
7
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2answers
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Are the creation and annihilation operators time-dependent?

Something that always confused me when first hearing about second quantization were the dependencies of the creation and annihilation operators. On the one hand I have seen expressions such as $$ \...
7
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2answers
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Can quantum fields be viewed as superpositions of classical fields?

At the end of my undergraduate quantum mechanics class, we looked at phonons. You can let $x_i$ be the position operator of an nth quantum harmonic oscillator, and couple the harmonic oscillators with ...
7
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1answer
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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 ...
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3answers
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Meaning of Fock Space

In a book, it says, Fock space is defined as the direct sum of all $n$-body Hilbert Space: $$F=H^0\bigoplus H^1\bigoplus ... \bigoplus H^N$$ Does it mean that it is just "collecting"/"adding" all ...
6
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2answers
884 views

Basis transformation of creation and annihilation operators

I am reading through the chapter on second quantization in Advanced Quantum Mechanics by Schwabl. In §1.5.1, the book suggests that because the two orthonormal basis $\{|\lambda\rangle\}_\lambda$ and $...
6
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4answers
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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 ...
6
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2answers
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Composition of squeeze operators?

I'm wondering if it exists a composition law for the squeezing operation ? I guess so for geometric reason, since they are (generalized, and the phase is annoying of course) hyperbolic rotations of ...
6
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1answer
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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 ...
6
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1answer
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Bogoliubov-de-Gennes (BdG) formalism

Suppose you treat the mean-field BCS superconductor Hamiltonian $H$ in "BdG style" by re-writing it as $H = \frac{1}{2} \sum_k \psi_k^{\dagger} H_{BdG} \psi_k$ where, in terms of original ...
6
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1answer
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Derivation of Rashba spin-orbit coupling in tight-binding model

Rashba spin-orbit coupling Hamiltonian in free space can be written as: $H_{\text{so}}=\int d^3r \Psi^{\dagger}(\mathbf{r}) \gamma (p_{x}\sigma _{y}-p_{y}\sigma _{x})\Psi(\mathbf{r})$. I expand $\...
6
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1answer
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Computing the density operator commutation relations (Atland & Simons)

I'm trying to work through Altland and Simons' example of interacting fermions in one dimension. It's in chapter 2, page 70 (you can find it here). They define fermionic operators $$ a_{sk}^\dagger $$...
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Why must the Bogoliubov transform preserve anticommutation relations?

$\mathbf{Background}$: In my research I am studying the Ising model, expressed in terms of Jordan-Wigner fermions: $$ H = \sum_{j=1}^n(c_j - c_j^\dagger)(c_{j+1} + c_{j+1}^\dagger) + \lambda c_jc_j^\...
5
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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{...
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3answers
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Schrödinger field operators and their commutation relations

I've got several questions regarding the so called second quantization of the Schrödinger equation. My professor introduced the field operators for the Schrödinger field by simply stating them as ...
5
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1answer
736 views

Naive quantization of Schrödinger field

I just started learning QFT and I was wondering if one is able to quantize the Schrödinger field similar to the way one is able to quantize electromagnetic or elastic mechanical wave modes. E.g. ...
5
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2answers
294 views

Is there a natural operator that is canonically conjugate to the Hamiltonian?

As is well known, the Heisenberg uncertainty principle states that the position and momentum satisfy an uncertainty relation, which follows from the canonical commutation relation \begin{equation} [\...
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2answers
363 views

An exactly solvable model of 2D Majorana zero modes

The Kitaev's Majorana Model is an exactly solvable model of p-wave superconductor with localized Majorana zero modes in 1D quantum wire. For the 2D case, the general theory of Majorana zero modes near ...
5
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2answers
265 views

Creation/Annihilation Operators and Positive/Negative Exponentials

One of the principal concepts in QFT is to consider the expasion of the field $$\phi(x)=\int{\frac{d^3 \vec{p}}{2(2\pi)^3\omega_\vec{p}}}(a(\vec{p})e^{-ipx}+b^{\dagger}(\vec{p})e^{+ipx}),$$ with ...
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2answers
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What is the physical interpretation of a field operator

So far in our lecture we defined creation operators $a^{\dagger}_{n}$ in the following way, that we said: Somebody got you a antisymmetric or symmetric N- particle state and now $a^{\dagger}_{n}$ ...