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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How to compute the Feynman propagator for the Proca field?
I was repeating each step of the exercise 6.4 of the Greiner's book "Field quantization" when I discovered that there is a passage which I can't reproduce, the calculations are lengthy and ...
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Why do we need the Dirac's hole picture?
When I have to quantize a Dirac field I have to start by the usual classical Lagrangian and find the associated Lagrange equations, then quantize the solutions promoting them to quantum operators. In ...
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What is the ground state of a Hamiltonian in $k$-space after Bogoliubov transformation? [duplicate]
Consider the following Hamiltonian in $k$-space, quadratic in terms of the $\gamma$ operators:
\begin{equation}
\hat{H}_2=\frac{1}{2}\sum_k
\begin{pmatrix}
\gamma_k^\dagger & \gamma_{-...
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Scalar Product Calculation and Identical Particles in Quantum Mechanics
In the book "Nolting, Theoretical Physics Part 5/2" (German), on Page 264, Formula 8.80, the author introduces second quantization in the case of identical particles. One considers the ...
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Expectation of exponential of fermion bilinears
I'm trying to better understand free fermion systems. In particular, I'm hoping to learn which quantities are straightforward to calculate, and which are more complicated.
How can I calculate $$\...
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Quantization of non-relativistic complex scalar field
I found that the taking the non-relativistic limit of the Lagrgangian for complex scalar fields gives
$$\mathcal{L} = i\dot{\psi}\psi^* -\frac{1}{2m}\nabla\psi \nabla\psi^*.\tag{1}$$
Now, when we ...
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Does geometric quantization work for second quantization?
I have been studying geometric quantization, and was wondering if a similar method could be employed for the second quantization. I imagine such a setup would involve “going up a level;” our “phase ...
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Fourier and inverse Fourier transformation of creation and annihilation operators
I've seen two versions of the Fourier transform for the creation and annihilation operators in 2nd quantization, where the annihilation operator of the Bloch/Wannier state is treated as the F.T./I.F.T....
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Anticommutator Relation of Quantized Fermionic Field and Fermi–Dirac statistics: How are these related?
I'm reading the Wikipedia article about Fermionic field and have some troubles to understand the meaning following phrase:
We impose an anticommutator relation (as opposed to a commutation relation ...
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Writing a two-body operator in second quantization
I have the following two-body operator
$$ \sum_{1\leq i<j\leq N} w(x_i-x_j)$$
acting on the Hilbert space $\mathcal{H}^-_N=\bigwedge^NL^2(\Lambda)$, which describes the interaction between $N$ ...
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What is the Hilbert space of a single photon?
I'm trying to understand the second quantization of photons. Following expression (4.3.5) in the lecture note:Second Quantization, a Hilbert space of a indistinguishable multiparticle system in second ...
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Coadjoint orbit method (geometric quantization) for second quantisation of free relativistic particles
The quantum theory for relativistic particles can be obtained by constructing the (projective) unitary irreducible representations associated to the integral coadjoint orbits of the Poincare group.
...
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How phonon emerges from the quantum mechanics of the lattice?
In all textbooks and lecture notes I've seen so far, a phonon is introduced by imposing the (second) quantization condition on the classical Hamiltonian of the bodies connected with springs.
However, ...
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Rewriting two-body operator in second-quantized form
I would like to understand the following identity for fermion field operators:
$$\psi^\dagger(x) \psi^\dagger(y) \psi(y) \psi(x) = \psi^\dagger(x) \psi(x) \psi^\dagger(y) \psi(y) - \delta(x - y) \psi^\...
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Vacuum state of Bogoliubov quasi-particles (continued)
This question focus on another aspect of my previous question. Consider a toy bilinear Hamiltonian consisting of two bosons $\{b_i\}_{i=1}^2$:
$$
\begin{align*}
\mathsf{H}[b^\dagger,b]
&= ...
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Matrix elements in second quantisation formalism
In a system with two orbitals $c$ and $d$ (each with two spin degrees of freedom), consider the Hamiltonian $$H=V(d^{\dagger}_{\uparrow} c_{\uparrow} + c^{\dagger}_{\uparrow}d_{\uparrow}+d^{\dagger}_{\...
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The underlying cause of ill-defined loop-integrals in Quantum Field Theory
One of the main causes which leads to ill-defined loop integrals in Quantum Field Theory is that the variables of a Field Theory, $\varphi(x)$ for instance, are Quantum Fields which are governed by ...
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Are two localized single-photon states always invariant under the particle exchange?
In a text book for quantum communication, I learned that one generates optical pulses (wavepackets), each of which contains only one photon. For instance, the state of two wavepackets are described by ...
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Current operator from derivative of Hamiltonian with respect to the gauge field
For the second-quantized Hamiltonian
$$
H = \int d^3x \psi^\dagger(x)\frac{(-i\nabla - q\vec{A}/c)^2}{2m} \psi(x) + H_{int},\tag{1}
$$
one can conveniently compute the current operator by
$$
j(x) = -c ...
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How to normalize the states in the continuous limit?
In quantum field theories we can perform the continuous limit, where we take the limit $V\rightarrow\infty$ for the volume system. In quantum optics, we can start by absorbing a factor $\left(\frac{L}{...
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Where did the the $\delta(0)$ go? [closed]
I am reading my notes for a lecture on quantum field theory. For some example of a 1d system with spatial coordinate $x$ and for a wave with frequency $\omega$, it starts with defining field operators ...
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Existence and uniqueness of vacuum of fermion or boson operators
Suppose I have a set of boson (or fermion) annihilation operators $\{a_i\}$ defined on a Hilbert space. These operators satisfy the canonical (anti-)commutation rules
$$
\text{boson:} \quad [a_i, a^\...
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Ground state of Bogoliubov quasi-particles (simpler version)
This is a simplified version of one of my previous questions. Let $b_1, b_2$ be two boson operators; their vacuum is denoted as $|0\rangle$, i.e. $b_i |0\rangle = 0$. We can make a canonical ...
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Reference request: Non-relativistic scattering in second quantization
There are at least two possible ways to go about computing the amplitude for $2\to2$ scattering of indistinguishable particles in non-relativistic quantum mechanics. The first is the method we all ...
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String theory and string fields
Currently the proven theory is the quantum field theory. This theory defines fields in "all spacetime" and particles are disturbances in these fields. These particles are punctual and ...
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Are photons actually particles at all?
I just read this answer to "What exactly is a Photon?" which has me a bit confused. It seems to be arguing that "photon" is just a catch-all term for any sort of interaction with ...
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Mathematically, what's the difference between the EM field in classical E&M and the EM field in QED?
Perhaps another way to put it is, what exactly does it mean to quantize the EM field and why is it necessary? What mathematical properties does the quantized version of the field have that the ...
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Defining normal distribution in canonical quantization
For simplicity, let us suppose quantized scalar field
$$\hat{\phi}=\int{\frac{d^3p}{\left(2\pi\right)^32E_\vec{p}}\left(a_\vec{p}e^{-ipx}+b^\dagger_\vec{p}e^{ipx}\right)}$$
How does one add a particle ...
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Second quantization interaction operator - what does the integration variable $\mathbf{r}$ represent?
In some texts (see [1],[2],[3]) the two particle interaction operator is defined as:
$$
V_{int.} =\frac 1 2\int d\mathbf{r}d\mathbf{r'} V(\mathbf{r},\mathbf{r')} \psi^\dagger(\mathbf{r})\psi^\dagger(\...
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Does particle creation and annihilation in QFT correspond to changing eigenvalues?
If I understood correctly from Griffiths' explanation of the ladder operators as applied to the quantum harmonic oscillator, the ladder operators represent increasing/decreasing the energy level of ...
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Is the Schrödinger equation incompatible with special relativity? If so, why?
In summary: If the SE doesn't work with SR, what specifically, in terms of the math, is the reason for this? I'm specifically interested in how this relates to the creation and annihilation ...
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Understanding quantized vector fields
Let $A^\nu$ be a 4-potential. In Folland's book on quantum field theory (page 117), he quantizes this object in the case where it represents a massive particle as:
$$A^\nu(x) = \int \sum_{j=1}^3 \frac{...
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What unitary operator achieves Bogoliubov transformation?
Let $a_1, \ldots, a_N$ be boson operators, $[a_i, a_j^\dagger ] = \delta_{ij}$. One often considers Bogoliubov transformation $a_i \to \sum_j (A_{ij} a_j + B_{ij} a_j^\dagger)$, where the matrices $A=(...
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Advantage of coherent path integral
I think(?) I am quite familiar with path integral over phase space, but not familiar with the coherent state path integral. What is the advantage of this coherent path integral besides the usual path ...
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Motivation behind introducing creation/annihilation operators into the Dirac equation
When studying the Klein-Gordon equation, the introduction of creation/annihilation operators was justified by recognizing a harmonic-oscillator-like equation which we know how to quantize. Is there a ...
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Is there a way to fix the ordering ambiguity of canonical quantization?
This question arose from my question on whether the vacuum energy is actually present for a free quantum scalar field
What is the right way to treat the vacuum energy?
Part of this discussion is that ...
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Creation and destruction of particles in the Path integral
Reading Zee's QFT book he defines the interacting partition function as follows
$$Z(J,\lambda)=\int\mathcal{D}\varphi \exp{\left(i\int d^4x \left[\frac{1}{2}((\partial\varphi)^2-m^2\phi^2)-\frac{\...
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Are there and constraints on the Mode functions for the Dirac equation?
I have some doubts about the following issue: conventionally, one expands the Dirac field operator in mode functions of positive/negative energy in order to introduce creation/annihilation operators. ...
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Hohenberg-Kohn Hamiltonian Expression
In Hohenberg and Kohn's paper on the inhomogeneous electron gas they express the Hamiltonian for "a collection of an arbitrary number of electrons moving under the influence of an external ...
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Commutation of kinetic energy operator with Hamiltonian
I am basically trying to calculate current energy operator $\hat{\mathbf{J}}_E(\mathbf{r})$ by using Heisenberg equation of motion as
$$
-\nabla\cdot \hat{\mathbf{J}}_E(\mathbf{r})=\frac{i}{\hbar}[H,\...
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The discrepancy between first and second quantised theories (Non-relativistic) in energy and position measurements
I realize I've asked a similar question before. In this question, I really want to focus on non-relativistic QM.
Energy and position measurements are straight-forward in the first quantised theory. ...
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How to evaluate the commutation relations in the Heisenberg equation of motion (Polarisation)
For a Hamiltonian in the following second quantisation form:\begin{equation}
H=\begin{aligned}
& \sum_{\vec{k}, s} E_{c, s}(\vec{k}) c_s^{\dagger}(\vec{k}) c_s(\vec{k})+\sum_{\vec{k}, s} E_{v, s}(\...
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In what way does QFT combine particles' wavefunctions to form fields?
If I understand correctly, in terms of the math of QFT, saying that particles are excitations in fields is mathematically equivalent to saying that a field is a representation of all possible ...
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Quasi-periodic motion of $N$-particle systems [closed]
My question is about the time evolution of multi-particle systems in QFT. There are such systems evolving a-periodically. I struggle with the treatment of them, always obtaining periodic or quasi-...
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Contradicting time reversal of complex order parameter
Consider a complex order parameter $\Delta$, e.g., in the superconducting case. It is well known that its time reversal would lead to $\Delta^*$.
On the other hand, we know the definition $\Delta\sim\...
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What does this notation mean in a many-body quantum system? [closed]
I'm studying the Many-body quantum system with the textbook Many-Body Quantum Theory in Condensed Matter Physics written by Henrik Bruus and Karsten Flensberg, and right now I'm feeling hard to ...
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What is the mathematical relationship between the wave functions of QM and the fields in QFT?
To be more specific, say I have the wavefunction for the position of some arbitrary particle, but then I also have the QFT operator (is that the correct term?) modeling the position of that same ...
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Hamiltonian of the BEC in 2nd quantization [closed]
If I have $N$ non-interacting particle (bosons) forming a BEC that is trapped to
$x = 0 $ (assume the system to be 1D) by an applied harmonic potential $V=\frac{1}{2}m\omega^{2}x^{2}$
How can I write ...
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Time reversal of superconducting order parameter?
This pedagogical material discusses the transformation of the superconducting order parameter $\Delta_k$ (matrix in spin space). Eq. (79) applies the time-reversal operator $K=-i\sigma^yC$ (complex ...
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Is the QED equation of motion for the wavefunction just the Dirac equation?
In the QED wiki article, they derive the equation of motion under the Lagrangian, using the classical Euler-Lagrange equation. This gives
$$(i\gamma^\mu\partial_\mu - m)\psi = e\gamma^\mu A_\mu\psi.$$...
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