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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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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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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73 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 ...
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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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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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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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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 ...
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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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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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Why is information indestructible in quantum mechanics? [duplicate]

Why is information indestructible in quantum mechanics?
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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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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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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 ...
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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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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 ...
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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}...
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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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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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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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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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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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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 $ ...
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Does Haags Theorem forbid Time-Evolution?

I didn't quite grasp the essence of Haags Theorem in the the way it is presented (for example on wikipedia), but the issue seems to be that if one wants to represent infinitely degrees of freedom ...
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Feynman propagator for Dirac fields and $i\epsilon$ prescription for analytic continuation

The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$...
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Optical theorem in QFT

I've been working with the Optical theorem in the case in which final and initial states are equals and I have the following doubt. Let's write the scattering matrix $S$ as: $$S = 1 + i·T \tag1$$ ...
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Why is it important that physics consists of unitary phenomena? [duplicate]

AFAIK all known quantum laws are unitary (except the collapse postulate, which is dubious anyway). Why is it important that the laws be unitary?
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Is Quantum Mechanics Compatible with Conservation of Information?

What is exactly the law of conservation of information? In quantum mechanics we have truly random outcomes in experiments, but doesn't this randomness mean that new information is produced and the law ...
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Quickly write a unitary transformation connecting two arbitrary states

This is for an introductory QM course. Say, I have pure states $\vert\alpha\rangle$ and $\vert\beta\rangle$. I would like to find a unitary operator that takes me from one to the other. What is the ...
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What is meant by unitary time evolution?

According to the time evolution the system changes its state the with the passage of time. Is there any difference between time evolution and unitary time evolution?
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Unitary CFT and Kac determinant

I am currently reading the BPZ paper, and trying to understand the nature of Minimal models. We compute the Kac determinant, and it is claimed that it must be positive for the theory to be a Unitary ...
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Why can't two different quantum states evolve into the same final state?

Is it true that two different states cannot evolve into the same final state? Can they achieve this state at different times? If yes, what is the proof?
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Anti-commutative hermitian operators

I have some trouble to prove the next statements: Let $A,B$ two anti-commutative hermitian operators, i.e. $\{A,B\}=AB+BA=0$. Does $A$ and $B$ share any eigenket?. If $U$ is an unitary ...
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Are there any black hole information loss solutions that do not resort to non-locality?

It seems as though all solutions to the information paradox resort to non local effects. What solutions do we have that preserve locality?
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What is the physical significance/importance of anti-unitary operators?

Time-reversal symmetry is an anti-unitary operator. I understand the mathematical definition of this, but what are the implications? What should/would one expect from anti-unitary operators? Are ...
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Use of Cutkosky rule, the Optical Theorem and Regge trajectories in pp scattering total cross-section calculation

Cutkosky rule states that: $$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$ putting $a=b=p$ in Cutkosky rule we deduce the ...
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Can someone explain why this matrix is unitary? [closed]

I have a matrix $$ \begin{bmatrix} a & b \\ e^{i\theta}b^* & -e^{i\theta}a^* \end{bmatrix} $$ Where $\theta$ is a real number, and $a$ and $b$ are complex number such that $|a|^2 + |...
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Unitary Representations in Conformal Field Theory

So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For ...
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Two Level Atom Rotating Frame

Introduction I am trying to work out the Rabi problem. In particular I am trying to work it out from a perspective focusing on the operators rather than the kets. Think Heisenberg picture rather than ...
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Analytic cotinuation between Minkowskian and Euclidean space, and causality

We can flip between Minkowkian and Euclidean signature by Wick rotation, and it is a well defined operation, provided there are no non - trivial singularities. Now, Unitarity in Minkowskian space ...
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If a theory violates unitary and renormalization of quantum theory, is it wrong?

If a theory of quantum gravity violates unitarity and renormalization at high energies, does that mean it is wrong and cannot be correct?
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Quick Question About Conformal Blocks

When a conformal block has dimensions and spin that violates its unitary bounds, does that make the block equal to zero. I'm asking because I'm trying to calculate 3D conformal blocks via a recursion ...
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Why can we use time-dependent perturbations when evaluating the S-matrix?

Suppose we have Hamiltonian $H_0 + V$. When working in the interaction picture we may derive the evolution operator of $|\psi_I(0)\rangle$ which is given by $$S(t,t_0) = T\left[\exp \left( -i \int_{...
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A question on the no-cloning theorem

When reading papers on the no-cloning theorem, I understand that one is searching for a unitary matrix $U$, such that for any state $\vert \phi_A \rangle$ in a Hilbert state $H_A$, we have that $U\...
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What effect would result from always keeping Page time >> Age of universe?

A thought experiment: I imagine, similar to early theories (eg: Hoyle), that the universe’s mass is not constant, but increases over cosmic time, in such a way that even the thinnest vacuum will ...
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What is the most likely/favoured theory put forth explaining the Black hole information paradox?

What is the most likely/favoured theory put forth explaining the Black hole information paradox? Is there any theory that is most favoured that has no flaws, or most likely explains how black holes ...