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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32 views

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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How can we introduce ghost fields as functions of complex variables [closed]

I have an equation of it which is written in the book string theory demystified by David MacMahon as b(z)c(w)=1/(z-w) How can I derive this equation. If someone is can help me.
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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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97 views

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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Quantization of the Gibbs distribution

Consider a simple quantum mechanical system, for example, the 1d harmonic oscillator. Given the inverse temperature $\beta$, the classical Gibbs distribution is the following function over the phase ...
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76 views

Quantum corrections in the phase space formulation

I'm trying to reconcile the following two statements: Quantum Mechanics gives physical predictions which are different than the predictions that are obtained in the $\hbar \rightarrow 0$ limit, that ...
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171 views

Planck-scale curvature in covariant LQG and quantization of length: does LQG apply also to the Planck-regime?

In the covariant approach of loop quantum gravity (see http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf ), the theory is defined on a "lattice", similar to lattice QCD. In this case ...
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Question about canonical quantization of the open string ghost system

In section 3.1.3 of Green, Schwarz and Witten book on superstrings, it is stated that the canonical anti commutation relations for the fermionic ghosts are $$ \{ b_{++}(\sigma, \tau), c^+(\sigma', \...
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43 views

Photon emmision from an accelerating particle

How does an accelerating charged particle emit a quantized photon? Quantization of light makes sense to me if we were talking about vibrating charged particles or electron orbitals. But what about a ...
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Is the "Push-Down" Quantization of Chern-Simons Theory part of a more general approach to Quantization?

I've recently started reading Axelrod, Della Pietra and Witten's original paper about the quantization of Chern-Simons theory. I'd like to know if the "push-down" quantization strategy they ...
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39 views

$\hat q \hat p$-quantization

I'm looking through the Berezin's paper 1971. And there are a couple of question that confuse me. It's clear why we need to use quantization procedure, because of the uncertainty principle and ...
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1answer
76 views

Hamilton's equation for generating functional

I've been reading E. S. Fradkin and G. A. Vilkovisky, “Quantization of Relativistic Systems with Constraints: Equivalence of Canonical and Covariant Formalisms in Quantum Theory of Gravitational Field....
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Discretized conserved values without necessarily using gauge symmetries

All of the examples I have seen for discrete conserved values (e.g. charge) invoke gauge symmetries, and thus extra degrees of freedom. Is it possible to have a discretized/quantized conserved values ...
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What is the third quantization and the creation and annihilation operators of universes?

We have only recently begun to undergo secondary quantization, and I know that for the introduction of the creation and annihilation operators, the existence of interacting quantum fields is necessary,...
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How to quantize a system if kinetic energy depends on coordinate?

In a standard physics course we usually learn that quantization of a system is ambiguous if momentum and position happen to be multiplied in the classical Hamiltonian (i.e. the classical Hamiltonian ...
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Why are commutators the first choice in describing observables that cannot be measured simultaneously?

In quantum mechanics, we convert Poisson brackets to commutators for the observables to account for the uncertainty principle. However, I do not understand why do we do this. What motivates us to ...
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Quantization of electromagnetic field: from free-space to media

When studying the quantization of the electromagnetic field, one seems to always derive everything for free space (no charges/currents). This involves solving Maxwell's equations to find modes (in ...
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Why are first class constraints harder to quantize than second class constraints?

I understand that the well known system with the second class constraints: \begin{align} &q_1 = 0 \\ &p_1 = 0 \end{align} has the apparent problem when performing quantization using the ...
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3answers
114 views

When a long wave photon is emitted by an electron, how come it is perfectly symetrical?

A long-wavelength, e.g. radio frequencies, of say, 1 km, has a period lasting about 1/300000th of a second. So for an imaginary fixed observer watching the incoming wave, it takes some time to go from ...
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Global mathematical structure of QFT

"Classical" gauge theories (e.g. electrodynamics combined with quantum mechanics) have the following global description: $A_{\mu}$ is a connection in a principle bundle The matter fields ...
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Some elementary relations in the quantization of a compact smooth symplectic manifold [closed]

In section 6.1 of https://arxiv.org/abs/1903.10792v1, there is a summary of relations in quantization on a compact symplectic manifold. These relations are as follows: $(M,\omega)$ compact $C^\infty$ ...
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121 views

Failure of canonical quantization of holonomic constraints

I am curious to know why canonical quantization fails for systems with holonomic constraints (dependent only on the position canonical variable). When googling, I notice that there is a lot of ...
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What is the rigorous definition of the verb "to quantize"?

I've studied QM and QFT for a couple of years now, so I'm familiar with the tersm "quantize", "quantization" and so on. I'm obviously also familiar with the Lagrangian description ...
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Difference between field-antifield and light-cone quantisation

I have learnt field-antifield quantisation and know that it can be used for very general gauge theories - open and reducible. I have not got much into light-cone quantisation but I am unable to see ...
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Mathematics for general gauge structure?

I am currently studying field-antifield (BV) quantization formulation from the review - arXiv:hep-th/9412228. This review gives a nice and quite involved treatment of general gauge structure and how ...
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Why secondary constraints in Quantum Theory?

I am self-studying dynamics of constrained systems and their quantisation from Rothe and Rothe book "classical and quantum dynamics of constrained systems". While using Dirac quantisation, ...
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1answer
95 views

How can a classical phase space be unquantizable?

On page 2 of the paper "2 + 1 dimensional gravity as an exactly soluble system" Witten claims that: Depending on its topology, a finite-dimensional phase space might be unquantizable, How ...
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Can one quantize Chern-Simons theory in the covariant phase space formalism?

The covariant phase space, which coincides with the space of solutions to the equations of motion, gives a notion of phase space which does not rely on a decomposition of spacetime of the form $M=\...
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1answer
56 views

Is it possible to minimize the number of axioms/rules of the canonical quantization?

In the standard canonical quantization procedure there are two rules. Transform all quantities to operators. Transform the Poisson bracket to a commutator. Of course it will be nicer to minimize the ...
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73 views

Polarization procedure in geometric quantization

The geometric quantization can be considered as an approach the formalize the way of associating a quantum theory corresponding to a given classical theory. Suppose we start with a sympetic manifold $(...
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1answer
158 views

Quantization and Commutation Relations

Why do we use commutation relations when quantizing any system? In the case of developing quantum mechanics from classical mechanics, we write the hamiltonian and then quantize it by having the ...
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WHY BRST formulation works: Conditions imposed on QFT to find (how many) BRST parameters

question: WHY BRST formulation works? In more details: What are the conditions we need to impose on QFT to find the BRST (global) symmetry? Why can we demand the BRST parameter $\epsilon$ directly ...
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Use the commutation relation to show that the conjugate momentum acts on eigenstates of $\hat{\Phi}$ as $ - i \delta / \delta\phi_a(\mathbf{x})$

This is part (b) of Schwartz's Problem 14.3 in his Quantum Field Theory and the Standard Model textbook. Suppose that we have a real scalar field operator $\hat{\Phi}(x^0,\mathbf{x})$ with conjugate ...
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140 views

Why is radial ordering necessary?

Suppose I have some conserved charge in a 2 dimensional CFT $$Q(|z|)=\int_{w=|z|}\text{d}w\,T(w).\tag{1}$$ The infinitesimal transformation induced on a field $\phi$ at $z$ is then $$[Q(|z|),\phi(z)]=\...
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What is the "secret " behind canonical quantization?

The way (and perhaps most students around the world) I was taught QM is very weird. There is no intuitive explanations or understanding. Instead we were given a recipe on how to quantize a classical ...
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Operator traces in Kontsevich quantization

In quantization, one studies maps from functions on the phase space to operators acting on the Hilbert space. Let's fix one such map and call it $Q$. Deformation quantization is based on the idea that ...
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957 views

Is this four-step recipe for quantization always valid?

I know that there is more than one way to go about quantization, but operationally, I find it useful to have a go-to set of steps that can convert a classical system to its quantum analog. Is there ...
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50 views

Quantization of an $\mathcal{c}$-algebra

I don't know if this is a valid question to ask, but I am wondering about the following: We are given a set of $\mathcal{c}$-number Lie-Brackets $$ [q_i,q_j] = 0= [p_i,p_j] \\ [q_i,p_j] = c_{ij}, $$ ...
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99 views

Quantization of $c$-number Dirac-Bracket

I have a question concerning the quantization of phase-space variables $(q_1, q_2, q_3, p_1, p_2, p_3)$ with the Hamiltonian $$ H = \frac{3}{2}(p_1^2+p_2^2 +p_3^2) $$ and the following non-commuting ...
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Do we understand why mass, charge or magnetic moment appear as quantized quantities?

There are two similar questions as the comment notified: 1. Reason for the discreteness arising in quantum mechanics? 2. How does quantization arise in quantum mechanics? I could see two points there: ...
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Validity of Canonical Quantization

I was studying about what does it mean canonical quantization treatment. But now I have the next question. Why if we establish canonical the commutation relations $$\left[q,p\right]=i\hbar,\quad \left[...
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149 views

Batalin-Vilkovisky quantization

Batalin-Vilkovisky (BV) quantization is way of quantizing a theory, which is apparently more powerful than BRST quantization. It has been used, for example, for string field theory, in the closed ...
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1answer
57 views

Fock Space representation for 1D Square Well Potential

I will make a few observations (if any of these is incorrect please let me know) and then ask my question :- i) For a Quantum Mechanical Harmonic Oscillator (QMHO) we have, at least, two kinds of ...
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Experimental verification of quantization

I understand quantization as a map from Symplectic Manifolds $M$ (either finite dimensional or not) to Hilbert Spaces $H$, along with a rule that attach to every function $F$ in $M$ a hermitian ...
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Gauge Symmetries as Redundancies vs Constraints

I am very confused by these two points of view. Consider a theory whose space of fields is $V$ and that has an action $S$. Thinking of a gauge symmetry as a redundancy is your description means that ...
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Why does a square root term make the quantisation of action difficult?

When going over my lecturer's notes on String Theory and trying to understand a particle as a theory of gravity in 1D, it is mentioned that the action $(1)$ is regularisation invariant, $$S=-m\...
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Name of concept: Replace classical variables by quantum operators

I feel like there was a name for this sleight of hand approach and I've been unsuccessfully trying to google it for a while. I think Heisenberg introduced it and it's basically "putting hats on ...
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Non-relativisitc Quantization of Classical Fields

When quantizing a theory of one particle, we are used to taking the classical dynamical variable $\gamma:\mathrm{time}\to\mathrm{space}$, the trajectory in time, and replacing it with another, ...
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On quantization of relativistic quantum theory on curved space

In his book "Lectures on quantum mechanics", at the end of chapter 3, Dirac states that "it does not seem possible to fulfill the conditions which are necessary for building up a relativistic ...

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