Field operators Definition and 22 Threads

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. Samama Fahim

    Total Momentum Operator for Klein Gordon Field

    As $$\hat{P_i} = \int d^3x T^0_i,$$ and $$T_i^0=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi-\delta_i^0\mathcal{L}=\frac{\partial\mathcal{L}}{\partial(\partial_0 \phi)}\partial_i\phi=\pi\partial_i\phi.$$ Therefore, $$\hat{P_i} = \int d^3x \pi\partial_i\phi.$$ However...
  2. S

    A Converting between field operators and harmonic oscillators

    Suppose we have a Hamiltonian containing a term of the form where ∂=d/dr and A(r) is a real function. I would like to study this with harmonic oscillator ladder operators. The naïve approach is to use where I have set ħ=1 so that This term is Hermitian because r and p both are.*...
  3. K

    I Quantum Field Operators for Bosons

    Consider the field creation operator ψ†(x) = ∫d3p ap†exp(-ip.x) My understanding is that this operator does not add particles from a particular momentum state. Rather it coherently (in-phase) adds a particle created from |0> expanded as a superposition of momentum eigenstates states...
  4. abivz

    I Obtaining the Dirac function from field operator commutation

    Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...
  5. abivz

    I QFT - Field operator commutation

    Hi everyone, I'm taking a QFT course this semester and we're studying from the Otto Nachtman: Texts and Monographs in Physics textbook, today our teacher asked us to get to the equation: [Φ(x,t),∂/∂tΦ(y,t)]=iZ∂3(x-y) But I am unsure of how to get to this, does anyone have any advice or any...
  6. K

    I Boundary terms for field operators

    Hello! In several of the derivations I read so far in my QFT books (M. Schawarz, Peskin and Schroeder) they use the fact that "we can safely assume that the fields die off at ##x=\pm \infty##" in order to drop boundary terms. I am not sure I understand this statement in terms of QFT. A field in...
  7. A

    I Does a Quantum Field Creation Operator Create Particles at a Given Location?

    Hi, It appears that the definition of a quantum field creation operator is given by $$\Psi^{\dagger}(\mathbf r) = \sum\limits_{\mathbf k} e^{-i\mathbf k\cdot \mathbf r} a^{\dagger}_{\mathbf k}.$$ But then if we examine how this operator acts on the vacuum state, we get $$\Psi^{\dagger}(\mathbf...
  8. P

    I Field operators and the uncertainty principle

    Hi, I am reading QFT by Lancaster and Blundell. In chapter 4 of the book the field operators are introduced: "Now, by making appropriate linear combinations of operators, specifically using Fourier sums, we can construct operators, called field operators, that create and annihilate particles...
  9. D

    A Time dependence of field operators

    In field theory we most of deal with theories whose Lagrangian densities are of the form (sticking to scalar fields for simplicity) $$\mathcal{L}= -\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m_{\phi}^{2}\phi^{2} + \cdots$$ where ##\partial := \frac{\partial}{\partial x^{\mu}}##...
  10. S

    A Commutation relations - field operators to ladder operators

    I would like to show that the commutation relations ##[a_{\vec{p}},a_{\vec{q}}]=[a_{\vec{p}}^{\dagger},a_{\vec{q}}^{\dagger}]=0## and ##[a_{\vec{p}},a_{\vec{q}}^{\dagger}]=(2\pi)^{3}\delta^{(3)}(\vec{p}-\vec{q})## imply the commutation relations...
  11. QuantumRose

    Commutator relations of field operators

    Here is the question: By using the equality (for boson) ---------------------------------------- (1) Prove that Background: Currently I'm learning things about second quantization in the book "Advanced Quantum Mechanics"(Franz Schwabl). Given the creation and annihilation operators(), define...
  12. D

    Interactions between field operators & locality in QFT

    Why is it required that interactions between fields must occur at single spacetime points in order for them to be local? For example, why must an interaction Lagrangian be of the form \mathcal{L}_{int}\sim (\phi(x))^{2} why can't one have a case where \mathcal{L}_{int}\sim\phi(x)\phi(y) where...
  13. Sigurdsson

    Difference wave mch. and field operators.

    Hi there I've recently started studying quantum field theory and I'm trying to understand the field operators. One thing that bugs me is the difference between field operators and wave mechanics operators. For instance, let's take the kinetic energy operator in wave mechanics for a single...
  14. S

    How Does Field Operator Evolution Hold in QFT?

    This is a doubt straight from Peskin, eq 2.43 ∅(x,t) = eiHt∅(x)e-iHt. This had been derived in Quantum Mechanics. How does this hold in the QFT framework? We don't have the simple Eψ=Hψ structure so this shouldn't directly hold. I'm sorry if this is too trivial
  15. D

    Problems on quantum field operators in QFT

    Hello! I met some annoying problems on quantum field operators in QFT.They are as follows: (1)The quantum field operator( scalar field operator, for example),is often noted as φ（r,t). Can it be interpreted as like this: φ（r,t) is the coordinate represetation of a...
  16. G

    Second quantization of field operators

    Homework Statement (from "Advanced Quantum Mechanics", by Franz Schwabl) Show, by verifying the relation \[n(\bold{x})|\phi\rangle = \delta(\bold{x}-\bold{x'})|\phi\rangle\], that the state \[|\phi\rangle = \psi^\dagger(\bold{x'})|0\rangle\] (\[|0\rangle =\]vacuum state) describes a...
  17. G

    What are the operations on vector field A and how do I simplify the results?

    Hello. I am stuck trying to find an understandable answer to this online: Carry out the following operations on the vector field A reducing the results to their simplest forms: a. (d/dx i + d/dy j + d/dz k) . (Ax i + Ay j + Ax k) b. (d/dx i + d/dy j + d/dz k) x (Ax i + Ay j + Ax k) I...
  18. P

    Field Operators in Klein-Gordon theory

    Currently I am working through a script concerning QFT. To introduce the concept of canonical filed quantisation one starts with the (complex valued) Klein-Gordon field. I think the conept of quantising fields is clear to me but I am not sure if one can claim that the equations of motion for the...
  19. E

    Field operators - how do they work?

    It seems to me that in the quantization of a classical field, one first takes the Fourier transform of the field to put it in frequency space: F \left(X, \omega \right) = \int f(X,t)e^\left(-i \omega t\right) then multiply by the annihilation and creation operators for a given wavelength: F...
  20. F

    Quantum Field Theory: Field Operators and Lorentz invariance

    [SOLVED] Quantum Field Theory: Field Operators and Lorentz invariance Hi there, I am currently working my way through a book an QFT (Aitchison/Hey) and am a bit stuck on an important step in the derivation of the Feynman Propagator. My problem is obviously that I am not a hard core expert...
  21. M

    Second Quantization and Field Operators

    When defining a field operator, textbooks usually say that one can define an operator which destroys (or creates) a particle at position r. What does this really mean? Are they actually referring to destroying (or creating) a state who has specific quantum numbers associated with the geometry...
  22. D

    Field operators in canonically transformed representations of the CCRs

    Here's a question about inequivalent representations of the CCRs... For a given Hilbert space representation, what is it that determines which set of field operators \phi(x), or \phi(f) if we want to get rigorous a la Wightman, gives us THE field operators for that representation. For example...
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