What is Canonical quantization: Definition and 18 Discussions

In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.
Historically, this was not quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization, and detailed it in his classic text. The word canonical arises from the Hamiltonian approach to classical mechanics, in which a system's dynamics is generated via canonical Poisson brackets, a structure which is only partially preserved in canonical quantization.
This method was further used in the context of quantum field theory by Paul Dirac, in his construction of quantum electrodynamics. In the field theory context, it is also called the second quantization of fields, in contrast to the semi-classical first quantization of single particles.

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  1. T

    I Prefactor in Canonical Quantization of Scalar Field

    Hey all, I am encountering an issue reconciling the choice of prefactors in the canonical quantization of the scalar field between Srednicki and Peskin's books. In Peskin's book (see equation (2.47)), there is a prefactor of ##\frac{1}{\sqrt{2E_{p}}}## whereas in Srednicki's book (see equation...
  2. K

    I Canonical quantization of Ashtekar's new variables

    Is Loop quantum gravity canonical quantization of Ashtekar's new variables correct ? if not in principle is there any particular ways to canonical quantization of Ashtekar's new variables ? are there other methods to quantization of Ashtekar's new variables ?
  3. Riotto

    A Canonical momentum ##\pi^\rho## of the electromagnetic field

    In David Tong's QFT notes (see http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf , page 131, Eq. 6.38) the expression for canonical momentum ##\pi^0## is given by ##\pi^0=-\partial_\rho A^\rho## while my calculation gives ##\pi^\rho=-\partial_0 A^\rho## so that ##\pi^0=-\partial_0 A^0##. Is it...
  4. maajdl

    A Canonical quantization of Electrodynamics: physical intuition ?

    Hello, I am freshly retired and enjoy going back to the fundamentals. I followed the wonderful courses by Alain Aspect on Coursera on Quantum Optics 1 and 2 . The quantization of Electrodynamics is really easy stuff. Just follow the correspondence between Poisson brakets and Commutators ... and...
  5. W

    Canonical Quantization: Steps to Find iħ

    Homework Statement For the canonically quantized operators, what are the step in between? how do you get the answer iħ? [q^,p^]=iħ q^ is the coordinate and p^ is the momentum.
  6. S

    I Canonical Quantization: Proving the Theory

    Hello! I read some books on QM and QFT but I didn't really noticed (or I missed it?) a proof for the canonical quantization. For example, for energy and momentum it makes sense to have opposite signs, due to Minkowski metric, be related to the variation of space and time, due to Noether theorem...
  7. K

    I Canonical Quantization: Choosing the Right Lagrangien

    Suppose we have a classical system described by a Lagrangien \mathscr{L}(x,t). The same system can be described by the Lagrangien \mathscr{L'}(x,t)=\mathscr{L}(x,t)+\frac{\mathrm{d}F(x,t)}{\mathrm{d}t}. where F(x,t) can be any function. If we now quantize the system by calculating the...
  8. G

    A Wave of an outgoing anti-particle in quantum field theory?

    Hello. I'm studying a course of the Quantum Field Theory and I got a question in a canonical quantization of a scalar field. I don't write a full expression of the field quantization here but the textbook said terms with ei(p⋅x - Ept) are associated with an incoming particle and terms with...
  9. H

    A Canonical quantization of scalar fields

    In the srednicki notes he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x) $$ to $$H = \int d^{3}p\frac{1}{2m}P^{2}\tilde{a}^{\dagger}(p)\tilde{a}(p) $$ Where $$\tilde{a}(p) = \int \frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}e^{-ipx}a(x)$$ Is this as simple as...
  10. S

    Canonical Quantization: Explaining the Covariance Paradox

    Hi all I am studying Quantum Field Theory. I read the following statement :"Canonical quantisation is intrinsically not relativistically covariant. Can anyone explain why?. Although everything we did from the beginning was lorentz invariant! Thank you .
  11. Z

    Canonical quantization of spin-3/2 particle

    If I want to calculate a decay of excited nucleons,sometimes I must treat the spin 3/2 field operator. If I use CG coefficient method, for example http://arxiv.org/abs/hep-ph/0210164 (page 3,the third equation) But it is a result,not the starting.The thought looks like a synthesis between...
  12. N

    Canonical Quantization problem; finding Schroedinget time-dependent equation

    Homework Statement A particle of mass m is confined in a Pösxhl-Teller potential as defined by: V(x) = -V0sech2(αx) Where V0 and α are constants representing the depth and width of the well. Use canonical quantisation to find the time-depndent Schrödunger equation for a particle in...
  13. C

    Scattering Amplitude in Canonical Quantization

    Hi there, I have little question: reading zee 2nd edition, I.8 (pag 64) i came up with this: start with <k_1 k_2| e^{-iHT}| k_3 k_4> and H=H_0 +u u=\lambda \int \phi^4 where H_0 is the usual hamiltonian for the free scalar field. Then, zee says that "expanding in \lambda, we obtain...
  14. L

    Canonical Quantization of KG field

    Hi, for my exam i"m re-reading Peskin&Schroeder and stumbled across equations 2.21-2.25 where the canonical quantization of the KG field is done. P&S start with doing a Fourier trf on \phi(x,t)=\int\frac{d^3p}{(2\pi)^3}e^{ip\cdot x}\phi(p,t) applying the KG operator in that results in...
  15. M

    Canonical Quantization: Dirac's Book & Gauge Theories

    Hi, I have recently been reading Dirac's book on Canonical Quantization of gauge theories, and I have a few questions: So in the quantization procedure we need to identify all the constraints in the theory. Once this has been done (if we are dealing with a gauge theory) we need to check...
  16. M

    Canonical quantization with constraints

    let be the Lagrangian (1/2)m( \dot x ^{2} + \dot y^{2}) - \lambda (x^{2}+y^{2}-R^{2}) with 'lambda' a Lagrange multiplier , and 'R' is the radius of an sphere. basically , this would be the movement of a particle in 2-d with the constraint that the particle must move on an sphere of...
  17. jal

    Spin foam model from canonical quantization

    The “math kids” are hard at work. http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3892v1.pdf Spin foam model from canonical quantization Sergei Alexandrov 26 may 2007 ----------- A quick search of dual 4-simplex found this supplementary information...
  18. I

    Question on Canonical Quantization

    This semester I took a course in Quantum Field Theory. It is difficult (the professor assumes you know everything) and I have so many questions... Starting with a lagrangian density, I was told that canonical quantisation is a procedure where we impose the usual commutation relation between...