# 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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### 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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### 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 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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### 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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### 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 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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### 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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### What is the precise formal correspondance between an oscillator and a quantum field?

A common route of introduction to quantum field theory is to note a similarity between the mathematical structure of a quantum harmonic oscillator and of a quantum field "at a point". The quantised ...
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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....
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### Dispersion relation in tight binding model with even indices only

Given a tight binding model with Hamiltonian $H= \sum_{i(even)}t[c_{i+1}^\dagger c_i+h.c]$ containing even indices only, how can we find out the dispersion relation? Attempt: My guess is that the ...
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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?
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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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### 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 ...
From what I understood from wikipedia, as well as some other resources, each phonon corresponds to a normal mode oscillation, and the creation operator to create a phonon of wavevector $k$ is: $$a^{\... 0answers 507 views ### Time ordering and Fermions Having time ordering operator for fermions, should it reverse sign if it swaps operators with opposite spin variable? In other words should T[c_{t_1,\uparrow}c_{t_2,\downarrow}^\dagger] return T[... 1answer 397 views ### Single particle operator in second quantization I want to understand why we write in the formalism of second quantization for a single particle operator $$\hat H=\sum_i \varepsilon_i \hat a_i^{\dagger} \hat a_i$$ where ... 2answers 564 views ### Annihilation and Creation Operators in QFT I have following question about creation and annihilation operators in QFT: The Klein-Gordon field is introduced as continuous interference of plane waves \mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})... 0answers 290 views ### A question regarding to the one-body operators in N-particle Hilbert space Recently, I've started studying the book "condensed matter field theory" by Alexander Altland and Ben Simons. At second chapter when he's trying to rationalize the one-body operators in Fock space, I ... 2answers 335 views ### How will the (anti)commutation relation between two different fermion fields look like? The anti-commutation relation between the components of a fermion field \psi is given by$$[\psi _\alpha(x),\psi_\beta^\dagger(y)]_+=\delta_{\alpha\beta}\delta^{(3)}(\textbf{x}-\textbf{y}). In ...
So my question is why do we need to write the coefficients $b$ (that after the second quantization are going to be promoted as the antiparticle creation operators) as complex conjugate? I mean, why ...