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 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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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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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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Electric and magnetic charge quantization: other derivations or perspectives?

We know (say from Griffiths E&M Problem 8.12) that the electric $q_e$ and magnetic charge $q_m$ (with a distance $\vec{z}$ apart) can store the angular momentum in the space: $$ \vec{L}=\int d^3 V ...
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1answer
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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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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
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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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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
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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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1answer
85 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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Center symmetry on wavefunction

Assume we know the wavefunctions of a SU(N) Chern-Simons (or YM) on a 3-mfld $M$, perhaps using holomorphic quantization. How do the center symmetry transformations act on the wavefunctions? Is it ...
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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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1answer
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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1answer
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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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1answer
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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
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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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1answer
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How do the electrons absorb energy in an discharge tube that is used for produce an emission spectrum?

When there's hydrogen in a discharged tube it produces an emission spectrum, emitting energy(photons). (Eg:-When an electron jumps from 3rd energy level to 1st energy level, the electron emits a ...
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Experimental tests of operator ordering

In quantization, we frequently run into ordering ambiguities. In general, this means that there can be inequivalent quantum theories corresponding to the same classical theory. Has there ever been an ...
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What do we mean by radial quantisation in CFT?

When we quantise QFT we do that in equal time slices. In CFT it is useful to use equal radius slices. Why is that the case? And what does it mean?
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General question about quantization procedure

As far as I understood, when we want to quantize a system, the procedure can be the following (but probably not the most general one): We start by writing down the Lagrangian of the system. We ...
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Planck's Hypothesis Derivation

Max Planck in 1900 used the quantisation of energy to explain black body radiation. Using what principles did he arrive at his final formula $E=nhv$?
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Why various molecules in an ideal gas at a particular temperature can have only quantized energies?

Why various molecules in an ideal gas at a particular temperature can have only quantized energies? Why can't they have the energies distributed in a continuous fashion? Following is an image taken ...
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1answer
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Quantising Classical Lagrangian

Suppose you have a system described by the following Lagrangian: $$L=(1-gq²)\dot{q}^2/2.$$ How would you quantize this theory? Do you need to symmetrize the Hamiltonian before promoting the ...
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Quantization of the Nambu bracket

The most simple quantum mechanical system consists of a canonical pair of operators $\{P, Q\}$ satisfying $$ P Q - Q P = i \hbar. $$ It is well known that there is a unique (modulo unitary maps) ...
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Can Newtonian gravity be quantized?

Today, nobody knows how GR is truly supposed to be married with QFT. As a result, the standard model as it is typically presented does not include gravity. Could it be modified to include Newtonian ...
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Einstein–Brillouin–Keller quantization rule, what does it really mean?

The Einstein–Brillouin–Keller method is a quantization rule going from classical mechanics to quantum mechanics, according to wikipedia: I have several question regarding the above description: what ...
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363 views

What's the difference between canonical quantization and second quantization?

I am wondering the difference between the canonical quantization and the second quantization in quantum field theory. For example, a harmonic chain, one can write down its lagrangian density $\...
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Single electron in conductive cavity

It is a basic result in electrostatics that a charge $q$ in an arbitrary cavity of an ideal conductor will generate a total charge $-q$ on the surface of the cavity in such a way that the electric ...
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Obstruction in quantization. Weyl Ordering

What is an obstruction in quantization? I've found that obstructions object of the study of a mathematical theory, previously concerned with homotopy. The problem is that to explain what an ...
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Creation and Annnihilator Operators: generality and meaning

I am studying my fisrst course in quantum mechanichs where we treated the example of the Harmonic Oscillator through the Weyl Heisenberg Spectrum Generating Algebra Method. In that context we ...

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