Questions tagged [unitarity]
In quantum mechanics, a unitary operator satisfies U<sup>†</sup> U = UU<sup>†</sup> = I, where † denotes Hermitean conjugation; such operators then specify Hilbert space automorphisms and preserve state norms, so then probability amplitudes and hence probabilities. Use for conservation of probability questions under unitary state transformations.
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Comparator operator in QFT
In Peskin and Shroeder, for a local $U(1)$ transformation, the comparator operator is expanded as:
\begin{equation}
U(x+\epsilon n, x) = 1 -ie\epsilon n^{\mu}A_{\mu} + \mathcal{O}(\epsilon^2)
\tag{15....
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Weyl- Squared Lagrangians
I'm studying conformal gravity theories, in particular I read that if we take $L=\sqrt{g}C_{abcd}C^{abcd}$ where $C$ is the Weyl tensor the theory we get is not unitary.
What does it means unitary at ...
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Is there uniqueness for field-operator and field-momentum operator, if I demand unitary equivalence for the field-operator?
This is a follow-up question to my question about the uniqueness of the field-momentum operator. The answer suggested that (partially) because the operators can act on different hilbert-spaces, there ...
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Can any “explicit time-dependence” of an observable in QM be also seen as a unitary transformation
My other question about plausibility of unitary time evolution in the Heisenberg-picture had me wondering:
If I can naturally argue the unitary time-evolution for any observable (that would be, the ...
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Why is time-evolution unitary - The Heisenberg-picture Version
There are various versions of this question already on this site, which attempt to justify / make plausible that the time evolution of quantum mechanical observables is unitary. Most of these ...
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$S$-matrix and in and out states
So, I have a short one. When observing scattering, we say that the amplitude for transition from one interacting state to some other interacting state same as this amplitude for free hamiltonian ...
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1answer
50 views
Unitarily reversing a projective measurement
We start with a particle in a pure superposition state. Let's say it is,
$$\vert\psi\rangle = \frac{1}{2}(\vert 0\rangle + \vert 1\rangle)$$
Alice sends this particle inside a box and the box ...
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2answers
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Can any linear but non-unitary “time-evolution operator” be normalized to a unitary one?
A comment to this answer to another question states
I would imagine that for any linear non-unitary time-evolution operator, I can find a unitary one that will yield the same expectation values for ...
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What is the black hole information paradox?
What is the black hole information paradox? My question is about the problem of whether information is lost when something falls into a black hole. Is information eternal? What about if a book falls ...
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Which unitary transformation should I use to change the frame reference properly?
I'm dealing with time-periodic Hamiltonian $H(t)=H(t+T)$ , where
$$ i\hbar \partial_t\psi(r,t)=H(r,t)\psi(r,t).$$
The periodicity lies on the potential (i.e. $V(t)=V(t+T)$ inside the $H(r,t)$).
The ...
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Gravity and complex numbers
In the lectures of Gary Gibbons on Supergravity held 2009@DAMTP http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf
it is remarkable that when he introduces spinors he ...
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Irrational Conformal Field Theory v.s. Non-Unitary Conformal Field Theory?
Unitary conformal field theories (CFTs) with irrational (or including the special case of rational) central charge is called irrational conformal field theory (ICFT).
Irrational conformal field ...
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Particle hole symmetry in 2nd quantization
In second quantization one the Particle hole trasnformation is defined as
\begin{align}
\hat{\mathcal{C}} \hat{\psi}_A \hat{\mathcal{C}}^{-1} &= \sum_B U^{*\dagger}_{A,B} \hat{\psi}^{\dagger}_B \\...
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1answer
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How Creation and Annihilation operator transform under an unitary transformation?
\begin{align}
\hat{\mathcal H}= \sum_{i,j} \hat{\psi}^{\dagger}_i H_{i,j}\hat{\psi}_j
\end{align}
The $\mathcal H$ is the full second quantized Hamiltonian for a system and $H$ is the single particle ...
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Could Time-Evolution be antiunitary?
There are serveral Arguments for Time-Evolution to be unitary, for example, time-evolution should preserve the norm of each given state (because elseways the probabillity Interpretation would not work)...
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Hamiltonian for a mode-shift operator
I have a discrete multi-level degree of freedom in my quantum system (for photons, for example this), which I write as $|l\rangle$. The degree of freedom is unbounded, i.e. $l$ can take ever positive ...
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Do all unitary operations manifest from time-evolution?
Let $|\psi\rangle$ be an element of a Hilbert space $\mathcal{H}$ and $U$ a unitary operator on $\mathcal{H}$. I am concerned with the actual physical manifestation of such a unitary operator in the ...
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Does there exists in physics an operator satisfies: $A^{-1}(t)=A(-t+ i \beta)$ , $\beta$ is a real number non-null?
Let $ t$ be a real number such that present the time. Really am interesting to know if there exists an operator satisfies the below property:
$$A^{-1}(t)=A(-t+ i \beta)$$ $\beta$ is a real number ...
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1answer
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Must an operator that preserves probability be unitary?
One property of the unitary operator is that it preserves the norm of the state-vectors:
$$
\langle \Psi | U^\dagger U | \Psi \rangle = \langle \Psi | \Psi \rangle
$$
If $U$ is unitary.
Is the ...
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1answer
50 views
How are anti-unitary transformations symmetric?
In the article on Wigner's theorem, unitary transformations ($U$) can be clearly seen as symmetric from:
$T: \Psi =\{e^{ia} \Psi|a \in R \} \mapsto \Psi^{'} =\{e^{ib}U \Psi|b \in R \} $
and hence,
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Can quDit gates be non-unitary
I have derived a quantum quDit gate that is implemented by photon Fock state inteference. It turns out to be non-unitary. I thought I must have made a mistake, so I have checked several times. I know ...
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1answer
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How can we prove that the Hamiltonian for any quantum system is Hermitian? [closed]
By applying partial time derivative to
$$\psi_t \rightarrow U \psi_{t_0}$$
we end up with an expression for the Hamiltonian
$$H = i\hbar\frac{\partial U_{t}}{\partial t}U^{\dagger}_{t}$$
where $U$ is ...
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Time-order evolution operator: Wei-Norman form and unitarity
I'm reading a paper[1] in which the propagator is calculated for this kind of Hamiltonian
\begin{align}
\hat{H}(t) = \omega(t)\hat{J}_3 + \Omega^*(t)\hat{J}_{+} + \Omega(t)\hat{J}_-.
\end{align}
...
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1answer
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Perturbative violation of the unitarity: what is it?
Consider the Fermi theory:
$$
\mathcal{L} = \frac{G_{F}}{2\sqrt{2}}\bar{n}\gamma_{\mu}(1-\gamma_{5})p \bar{\nu}\gamma^{\mu}(1-\gamma_{5})e
$$
The cross section of $2 \to 2$ scattering calculated ...
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40 views
Unitary equivalence of two representations
Suppose we make a transformation from Minkowski space to another coordinate system. What does it mean to say that the two spaces are unitarily equivalent? I have often seen the comment that if the ...
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1answer
91 views
Change of reference frame for a wavefunction: same modulus but different currents?
Suppose that, at a certain $t=0$, one has a wavefunction
$$
\psi=\psi(x,y)
$$
defined on a plane and well normalized to $1$. Coordinates (x,y) refer to the frame $xOy$.
How does the wavefunction ...
3
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1answer
76 views
Why is important for the energy density to be positive definite in field theories?
Why is important that the energy density be positive definite in field theories?
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1answer
59 views
Longitudinal polarization of gluons in loop
I have a short question about the possible gluon polarization in loop diagrams. For external gluons, we only want the 2 transverse polarizations.
In Peskin-Schroeder it is explained that in Feynman-...
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Generically, why do we want to evolve states with unitary operators? [duplicate]
Why is it so important that operators that evolve states are unitary?
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How did the book derive the following Unitary Operator expression in Mach Zehnder experiment?
I'm reading Renato Portugal's "Quantum Walks and Search Algorithms". In The Postulates of Quantum Mechanics, under the heading 'Evolution Postulate', there is the Mach Zehnder experiment, with the ...
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2answers
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What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time?
I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of ...
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2answers
183 views
QFT perturbation theory
I would like to clarify the following statement:
Perturbation theory (PT) in QFT is derived with several assumptions such as: adiabatic interaction, spectrum is bounded downward...
This statement ...
2
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2answers
95 views
Why does the $\phi$-cubed theory have no ground state?
In the book of Sredinicki's, he claimed that the $\phi^3$ theory has no ground state, hence this is not a physical theory.
My question is that I can't see why this system has no ground state. And I ...
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4answers
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Why is $\exp \left ( \frac{\pi}{2\hbar}(L_x^2 + L_y^2) \right )$ not a unitary operator? [closed]
I should prove that $$\exp \left ( \frac{\pi}{2\hbar^2}(L_x^2 + L_y^2) \right )$$ is not a unitary operator. Where $L$ is the total angular momentum of a 2-particle system ($L = L_A + L_B$ for the ...
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2answers
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Why is information indestructible in quantum mechanics? [duplicate]
Why is information indestructible in quantum mechanics?
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1answer
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Why quantum map must be hermitian?
Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $
the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$
Why $\mathcal{...
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1answer
122 views
How do we find a time-independent Hamiltonian that generates a given unitary transformation?
I know that for time independent Hamiltonians we can make the statement
$$U = e^{-iHt}\tag{1}$$
where $H$ is a time-independent Hamiltonian (divided by $\hbar$) and $U$ the unitary, also known as ...
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0answers
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Entanglement invariant under local basis change?
Today in the lecture the professor said that if we have an entangled state between two systems A and B
$$\mid \psi_{AB} \rangle = \frac{1}{\sqrt2}(\mid 00\rangle+\mid 11\rangle)$$
There is no local ...
2
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1answer
76 views
Does the wavefunction probabilities have to sum to 1? [duplicate]
In quantum mechanics we are often told that $\int |\psi(x,t)|^2 dx^3 =1$.
i.e. the probabilities have to sum to 1. And that this implies the time evolution operator is unitary.
But can't we define ...
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3answers
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Why is the information paradox restricted to black holes?
I am reading Hawking's "Brief answers". He complained that black holes destroy information (and was trying to find a way to avoid this). What I don't understand: Isn't deleting information quite a ...
2
votes
1answer
363 views
Why first-order Born Approximation doesn't satisfy optical theorem?
First-order Born Approximation in Quantum Mechanics states that scattering amplitude is a Fourier transform of potential:
$$
f(\theta) = \int d^3 r^{\prime} e^{-i (\bf k - k_i)r^{\prime}} V(r^{\prime}...
3
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1answer
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Wigner proof of the non-existence of finite unitary representation of the Lorentz group
I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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What's the problem with Black Hole evaporation?
Black hole evaporation is not unitary because it takes a pure state to a mixed state. On the other hand, ordinary decay processes in Quantum Mechanics do not seem very unitary either. (For example, if ...
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3answers
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Domains of $H$ and $U(t) = \exp(-iH t )$
I am not so familiar with functional analysis. But in my impression, the Hamiltonian $H$ is often not defined everywhere on the Hilbert space. On the other hand, the time evolution operator $U(t)$, ...
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Do longitudinal and scalar have anything to do with Faddeev-Popov ghosts?
In his this book, Hatfield calls ghosts the negative states appearing in the covariant (Gupta-Bleuler) quantization prescription of the electromagnetic field (page 89). When discussing Yang-Mills ...
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1answer
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Doubt in Weinberg's book on Quantum Field Theory
In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as:
\begin{equation}
U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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1answer
135 views
Why in quantum mechanics must orthogonal states stay orthogonal? [duplicate]
Given two states $|A(t)\rangle$ and $|B(t)\rangle$. If $\langle A(0)|B(0)\rangle=0$ then for all $t$, $\langle A(t)|B(t)\rangle=0$.
This is a fundamental rule of quantum mechanics. And we can imply ...
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2answers
495 views
Logarithm of Operators in Quantum Mechanics
In an operators algebra $\mathcal{A}$ one can consider a self-adjoint (i.e. real) operator $H$ and note that $$U=e^{iH}$$ exists and is unitary. A mathematical question will be whether any unitary ...
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1answer
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Why do we use infinitesimal forms of operators?
In many undergraduate texts on quantum mechanics (I'm using Modern Quantum Mechanics 2nd Edition by Sakurai as reference here), the treatment of angular momentum goes something along the lines of:
...
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2d CFT and first intersection of vanishing curves
In the search for 2d Unitary CFT's, we use the Kac determinant to find the null curves, and remove regions in parameter space(c and h space) where the determinant is negative.
Now, for $h>0$ and $ ...
