Questions tagged [quantization]
Quantization refers to the procedure or methodology for replacing a classical system by a quantum system. If the question is about the quantized or discrete behavior of a phenomenon use the [discrete] tag instead.
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Why can't we "simply" quantize Maxwell's equations without a Lagrangian to create a quantum theory of electrodynamics?
Useful quantum field theories like quantum electrodynamics (QED) suffer from a litany of problems related to the fact that, at least in their usual Lagrangian formulation, interactions between the ...
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Dirac procedure for Wheeler De Witt equation
After computing the Hamiltonian constraint and the momentum constraint in general relativity the Hamiltonian constraint is turned into an operator equation and solved in a manner similar to a ...
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Asking for explanation of Einstein's critique of the non-invariance of Bohr-Sommerfeld quantization
I am looking to understand better what problem might come from the claimed non-invariance of the Bohr-Sommerfeld quantization, which Einstein criticizes in his article On the Quantum Theorem of ...
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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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Naive first-quantization of the Dirac field?
To begin with, please note that I am fully aware of the differences between the confusingly named "first quantization" and "second quantization", and how they correspond to ...
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Loop Quantum Gravity vs Polymer Quantization
What is the linkage between Loop Quantum Gravity and the approach of the Polymer Quantization?
I know you get a lattice using the correct polymer representation, so that's a good toy model for the ...
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Deriving the ghost Lagrangian in Peskin and Schroeder
On page 514 of Peskin and Schroeder, the book derives
$$\tag{16.31} \det\bigg(\frac{1}{g}\partial^\mu D_\mu\bigg)=\int\mathcal{D}c\mathcal{D}\overline{c}\exp\bigg[i\int d^4x\overline{c}(-\partial^\mu ...
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Classical systems with compact phase space
In the Hamiltonian formalism of classical mechanics, a system with configuration space $Q$ is represented by a symplectic manifold $(T^*Q,\omega^\mathrm{can})$ called the phase space. The dynamics are ...
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Can we call it "quantization" when we specify Hilbert space and operators to write a classical field theory into a quantum theory?
Can we call it quantization when we specify Hilbert space and operators to write a classical field theory into a quantum theory? Suppose there is a single spin 1/2 system with Hamiltonian $\hat{H}=\...
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Integration measure for Polyakov's path integral
In section 3.4 of Blumenhagen's Basic Concepts in String Theory, where path integral quatization is presented, and we are given the partition function for the Polyakov's path integral
$$Z=\int \...
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Is Loop quantum gravity an unadulterated quantisation of general relativity, or does it have additional assumptions?
I was reading this Phys.SE answer written by user346. At the end of point 3, they say they've only made a change of canonical variables from the ADM formalism to get the Ashtekar formalism. Then point ...
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Using the EoM in the canonical quantization of EM field
Starting from the classical electromagnetic field, there two approaches to quantization that I want to compare. The problem arises when I write the classical fields in terms of $a$ and $a^*$, which ...
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Why is the vibration of chemical bonds quantized but the rotation about single bonds in molecules is not? [closed]
Vibration of bonds molecules is quantized. Rotation of entire molecules is quantized. Rotation about single bonds in molecules is not quantized.
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"Constrain then quantise" vs. "quantise then constrain"
Consider a classical system whose configuration space is a manifold $M$, and which is subject to some constraint $\mathcal{C}=0$.
[E.g. the system could be a particle moving in $M=\mathbb{R}^n$, with $...
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Does geometric quantization work for arbitrary "particle with constraint + potential" systems?
I was struck by the following line in Hall's Quantum Theory for Mathematicians (Ch. 23, p. 484):
In the case $N = T^*M$, for example, with the natural “vertical” polarization, geometric quantization ...
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When are two quantum descriptions / models equivalent?
I am occupied with different types of quantization methods for constrained systems. I start with a constrained phase space and then follow two different paths to get rid of the constraints. In the end,...
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Does $H=V(p) +x^2$ give a well-behaved Quantum Theory?
$V(p) $ is the Coulomb potential. I think this theory should be fine because Hamilton's equations are somewhat symmetric in $x$ and $p$. The momentum space Schrodinger equation will be:
$$i\frac{d\psi ...
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Is there a both manifestly covariant and unitary formalism of Quantum Field Theory?
The Lagrangian formalism is only manifestly covariant and the Hamiltonian formalism is only manifestly unitary.
In classical field theory, there exists the De Donder Weyl formalism, which is ...
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Light-cone quantization of open string as derived in Polchinski
Polchinski uses the following gauge conditions, but I don't follow this procedure of gauge fixing and quantization:
\begin{align}
X^+ = \tau, \tag{1.3.8a} \\
\partial_\sigma \gamma_{\sigma \sigma} = 0,...
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What if we skip polarization in geometric quantization?
In QM, we most frequently work with "position-space" representation of the CCR
$$
\mathcal{H} = L_2(\mathbb{R}, dx), \quad
X = x, \quad
P = - i \hbar \frac{d}{dx}.
$$
Sometimes it's useful ...
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Constructing a field theory action for the point particle in curved space
The point particle action in the Hamiltonian formalism is
$$
S = \int d\tau \Big( -p_\mu \dot{x}^\mu - \frac{e}{2}(g^{\mu\nu} p_\mu p_\nu - m^2) \Big) \ ,\tag{1}
$$
where I explicitly displayed the ...
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De Broglie's hypothesis for and resulting reasons for the lack of absorption of photons not at the specific discrete energy levels required
Essentially what the title is.
Although I know de Broglie's standing wave model for electron orbits has problems, I have a question regarding why atoms will not absorb photons not of the specific ...
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Is the expectation value of an operator obtained by canonical quantisation always the classical value? [closed]
I understand that generally, for some Hermitian operator $\hat{A}$, the classically measured value of a system is given by
\begin{align}
\langle \hat{A}\rangle=\langle\psi| \hat{A}|\psi\rangle
\end{...
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Mode expansion for $p$-branes
In the quantization of $p$-branes, for $p>1$ what is the mode expansion? What I am after, more specifically, is: what is the configuration around which the expansion is made? For example, for the ...
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Why does the Dirac Lagrangian not already use operators (instead of canonical quantization)?
I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations ...
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Derivation of Equation 10.97 in Zettili: Quantization of the EM Field
I am reading through Zettili, and I am stuck on one of the steps that Zetilli makes when quantizing the EM field, specifically with page 567.
Zettili first introduces the Fourier series for the ...
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Path integral quantization of the EM field in Peskin and schroeder
I'm studying path integral quantization of the electromagnetic field using Peskin and Schroeder secdtion 9.4. We want to compute the functional integral
$$\tag{9.50} \int \mathcal{D}A\,e^{iS[A]}.$$
We ...
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Operator Ordering Conventions and Symmetry
Quantization procedures may need operator ordering conventions to avoid ambiguity. In classical theories, classical observables are often described by smooth functions, so the order of observable ...
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Mathematical equivalent of Fundamental nature of charge [closed]
How to mathematically represent the fact that electric charge is a fundamental quantity? i.e. that it cannot be explained in terms of other things, for example, the normal force can be explained as ...
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The equivalence principle and the existence first quantization?
So imagine we are doing Einstein's famous thought experiment where we are locked in an elevator so narrow we are unable to detect tidal forces. The equivalence principle suggests that we should be ...
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First and second quantization of relativistic mechanics
Classical mechanics can be written in a lagrangian formalism. If one quantizes this theory, we get quantum mechanics. Let us continue this process:
Relativistic mechanics can also be written in a ...
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Doubt on the geometry of "quantum phase space"
In Jose & Saletan's "Classical Dynamics", they show the global structure of Hamiltonian mechanics: you then have a $Q$ manifold (configuration space), and the phase space structure is ...
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Wave equation for lightcone coordinate $X^-$
A quick question from Polchinski volume.1 : He claims in p.20 that the worldsheet lightcone coordinates $X^\pm$ also (i.e. in addition to the transverse coordinates $X^i$) satisfy the wave-equation. ...
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What does it mean to "quantise" a system?
Suppose we have a physical system, let's say a ring of $N$ atoms held together by elastic force. (This is just an example, we could have picked any physical system)
Classically we can easily find the ...
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Are there quantum gravity theories in which spacetime itself is regarded as quantum in nature?
In quantum gravity, it's tried to quantize the gravitation. However, if I got it correctly, most quantum gravity approaches try only to quantize gravity as a force, the curvature of spacetime, not the ...
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Confusion about the definition of negative angles between two D-branes
I have a conceptual question regarding a paper I am reading. More precisely, the first appendix starting on page 29.
In this paper, schematically, the classical oriented open strings get quantised ...
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What physical observables are the creation/annihilation operators for the EM field made from?
The explanations of quantizing the Electric/Magnetic (E/M) fields that I've read have all basically worked by using the Coulomb gauge in free space to define the vector potential in some volume as
$$
\...
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Basic question in similarities and difference on quantizations
In physics, usually quantization means canonical quantization. i.e., which we treat classical objects to quantum operators. i.e., For the association $Q:f \mapsto \hat{f}$ from functions on the ...
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Momentum operator in Geometric Quantization vs momentum operator on arbitrary curved space(time)s
In the following stack exchange post Momentum Operator in curved spacetime (QFT) a general expression for the momentum operator is given for a Riemannian manifold $(M,g)$. Similarly, Frederic Schuller'...
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Physical motivation of quantization
I am a mathematics student recently looking into (geometric and deformation) quantization. I'd like to know more about their physical motivations. Here by "quantization" I mean any process ...
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About Loop Quantum Gravity and concerns with its "polymer" quantization. Has it ever been addressed or answered/justified?
Referring to Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything
Underlying papers are: J. W. Barrett, “Holonomy and path structures in general relativity and ...
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The different frameworks around the string
I am studying string theory and I realize that the relations between the different frameworks are not clear to me. Following this question, one could repeat the discussion but now taking $p=1$. We get ...
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Quantum field theory not related to classical field theory [duplicate]
Overvation 1: Whatever quantization process is used, it is common to define a QFT from a classical field theory.
Observation 2: On the other hand, given a lagrangian QFT, one could try to "...
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Geometric Quantization of Dirac spinor in QFT
I have been using resources such as, Geometric quantization, Baykara Uchicago, to get a deeper insight into geometric quantization. However, it seems to me that this theory is only valid for quantum ...
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Order of product of $x$ and $p$ while deriving Hamiltonian from a Lagrangian in Quantum Mechanics
Everyone who has taken a course in Quantum Mechanics has at some point derived a quantum Hamiltonian from a Lagrangian. However, I can't seem to find any reference on the topic.
My question is ...
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Does it make sense to quantize perfect fluid?
Wikipedia (see here) says perfect fluid may be quantized. I do find an article (arXiv 1011.6396) about this, and the procedure is straight forward. What I do not understand is whether this ...
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Discrepancy between the two different equations of the momentum operator
i am doing a thesis on the quantization of a real scalar field in a gravitational wave background. I am doing this in lightcone coordinates, so $u$ is $z-t$. I start with an action and define a ...
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When can we quantize the Hamiltonian for an LC circuit?
For a superconducting qubit, we start with an LC circuit and "quantize" it, mapping the variables analogously to the variables for the harmonic oscillator.
In general, when are we allowed to ...
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A Hamiltonian with a potential depending on the momentum
Imagine we have a Hamiltonian, whose potential depends on velocities (and hence on the momentum),
like, for example,
$$ H= \frac{p^{2}}{2m}+ V(x,p)$$
then how can I quantize that?
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Function of noncommutative operators: how should the powers in its Taylor expansion be arranged, and how to take partial derivatives?
Let $F:\mathbb R ^n\to\mathbb R$ be a function that has a Taylor expansion, then it can be written (expanded at $a$) as
$$
F(x)=\sum_{\alpha} \frac{(x_1 - a_1)^{\alpha_1}\dots(x_n - a_n)^{\alpha_n}}{\...
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