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Results tagged with operators
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In physics, an operator is almost always either a square matrix or a linear mapping between two function spaces (defined on, say, $\mathbb R^n$). Operators serve as observables and as time evolution operators in Quantum Mechanics. This tag will most often find valid use in quantum mechanics; don't use this tag just because your equations contain "everyday operations" like $\times$, $+$!
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How are Fermionic and vector valued quantum field theories rigorously defined?
A scalar (spin 0) quantum field is rigorously defined as an operator valued distribution. By Wick rotating to Euclidean space we can view a quantum field theory as a measure over distributions. How mu …
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How are Schwinger functions defined as moments if they are actually operators?
But in rigorous approaches to quantum field theory the quantum fields are taken to be operator valued distributions, so after fixing test functions the integrand in (1) are all operators. …
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Equivalent definitions of Wick ordering
Let $\phi$ denote a field consisting of creation and annihilation operators. … before annihilation operators. …
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Defining Wick/normal ordering beyond rearranging the order of annihilation and creation oper... [duplicate]
Most introductory quantum field theory books define Wick ordering as rearranging a product of creation and annihilation operators such that all the creation operators appear to the left of any annihilation … operators. …
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Motivation behind introducing creation/annihilation operators into the Dirac equation
When studying the Klein-Gordon equation, the introduction of creation/annihilation operators was justified by recognizing a harmonic-oscillator-like equation which we know how to quantize. … Is there a similar justification when introducing these operators for the Dirac equation? Most of the resources I have looked at simply state them and move on. …
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How does rotation appear in the components of the angular momentum operator?
.$$
Promoting the right hand side to operators:
$$X_i \rightarrow x_i\\
P_i \rightarrow -i\hbar\frac{\partial}{\partial x_i}$$
I would expect $\hat{J}_3$ to be
$$(\hat{J}_1\psi)(x) = -i\hbar\Big(x_1\frac …
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Why is the wave function of a particle with definite momentum $p$ given as $e^{ipx/\hbar}$?
In many textbooks it is stated that, in position space, the wave function of a particle with definite momentum $p$ is given by $e^{ipx/\hbar}$. I know that the $\hbar$ comes from the de Broglie hypoth …
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Calculating the contraction of a field operator and a creation operator
In Folland's Quantum Field Theory (section 6.4) he considers a field:
$$\phi_\pi = \sum_\tau \int f(\textbf{q})\big[u(\textbf{q}, \tau, \pi)a(\textbf{q}, \tau, \pi) e^{-iq_\mu x^\mu} + v(\textbf{q}, …
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How do we know that all quantum fields are Fourier transforms of creation and annihilation o...
I must have missed it while reading the text, but how do we know that all quantum fields
Involve only creation and annihilation operators and not any other kind of operator. … Are Fourier integrals of creation and annihilation operators. Where does this come from? (Why do we take the Fourier transform?) …
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How does the Klein-Gordon equation represent a classical field? [duplicate]
This is what I know about the Klein-Gordon equation so far. Suppose we are working with natural units such that $c = 1$. Then we may obtain the Klein-Gordon equation by considering any 4-vector $p^\mu …
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How to interpret quantum fields?
As an analogy of what I am looking for, suppose $f(x,t)$ represents a classical field. Then we may interpret this as saying at position $x$ and time $t$ the field takes on a value $f(x,t)$.
In quantum …
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What does it mean to apply a creation or annihilation operator to a free field, e.g. $\langl...
So far, I have studied free fields and some basic computations involving them, such as creation and annihilation operators. … Hence if $a$, $a^\dagger$ respectively, represent the annihilation and creation operators, then $a\varphi$ lowers the number of particles in $\varphi$ by 1 and similarly for the creation operator. …
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How do we know that the operator $U(t) = e^{-itH/\hbar}$ corresponds to time?
By Stone's theorem on one-parameter unitary groups we know that there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators. …
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Why do we care about the canonical commutation relations?
Suppose $\hat{x}$ and $\hat{p}$ are the position and momentum operators, it can be shown that
$$[\hat{x}, \hat{p}] = i\hbar\mathbb{I}.$$
The Stone-von Neumann theorem tells us that that the above is unique … My current interpretation of commutators is, informally speaking, that they measure the extent to which two operators commute. …
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Why does QFT require operator-valued distributions? [duplicate]
For example, the loss of determinism going from classical mechanics to quantum mechanics is (at least from what I understand) the motivation for upgrading observables from scalar-valued maps to operators … What is the analogue for upgrading operators to operator-valued distributions? …
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