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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Matrix for $\pi/2$ pulse?

If we have a two-state system \[ \newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\...
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Do all unitary-preserving regulators necessarily turn real loop integrals into pure imaginary numbers?

The optical theorem, which results from the unitarity of the $S$-matrix, relates the imaginary part of the forward scattering amplitude to the total cross section. When using this theorem in practice, ...
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Perturbative proof of unitarity of $S$-matrix in QED

In any standard textbook on QFT I know it is claimed that the $S$-matrix in QED is a unitary operator. I have never seen any proof of it. This should be compared with the analogous property of $S$-...
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Why is $SU(3)$ chosen as the gauge group in QCD?

Why is $SU(3)$ chosen as the gauge group. Why not $U(3)$? Why does it even have to be unitary?
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Stone's theorem on one-parameter unitary groups and new self-adjoint operators

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position, ...
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What are the theoretical / mathematical problems in discarding negative solutions of Dirac equation?

I read some Q&A about it, but my question is why Dirac was so sure that he could not discard negative energy solutions. It seems so natural that energy must be positive, that I suppose that if ...
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Proof of normalizing the wave function

So suppose we have the wave function $\phi(x,t)$ in the context of quantum mechanics, that satisfies the Schrodinger's equation. We want to see that if we normalize this function at $t=0$ then it will ...
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Anti-commutator for annihilation and creation operators: ordering of indices

I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
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Do black holes comply with the principle of unitary evolution?

Claus Kiefer, "Quantum gravity", 3rd ed., page 220/221, says in chapter 7 "Quantization of black holes": "A theory of quantum gravity should give a definite answer to the question of whether ...
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Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices

This is exercise 4.38 from Nielsen and Chuang: Prove that there exists a $d \times d$ unitary matrix $U$ which cannot be decomposed as a product of fewer than $d-1$ two-level unitary matrices. If ...
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Calculating a different partition function of E&M leads to wrong results?

Consider pure a $U(1)$ Yang-Mills action $$S=-\frac{1}{4g^2}\int d^4xF^{\mu\nu}F_{\mu\nu}$$ Usually, one calculates the corresponding partition function $\mathcal{Z}(J^{\mu})$ by adding some external ...
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How to check whether Weyl field Hamiltonian is bounded below?

When constructing the Lagrangian for a two-component left-handed Weyl field $\psi$, in e.g. Srednicki, one rejects the choice of $\partial^\mu \psi \partial_\mu \psi+\partial^\mu \psi^\dagger \...
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How are unitary representations different from other representations?

I understand that unitary representations arise naturally in quantum mechanics when groups act on the Hilbert space in a way that preserves probability. I don't understand what details make unitary ...
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Are all representations of a finite group unitary?

I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96): Unitary representations The all-important unitarity theorem states that finite ...
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Assumptions in the proofs for the optimality of Grover's Search Algorithm

I was trying to understand this paper in which it is proved that Grover's search algorithm is optimal. On page 4, beginning of section 2 of the paper the author says the following In the proof I ...
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Importance of Hamiltonian being bounded from below

In perturbation theory one doesn't really explore much of the global properties of Hamiltonian. So I think PT is not affected by whether a Hamiltonian is bounded below. Yet in the linear potential or ...
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Is every unitary operator induced by a Hamiltonian?

Diving deeper into the mathematical inner workings of quantum mechanics: The set of unitary operators on the Hilbert space $\mathcal{H}$ forms a group. While for finite-dimensional Hilbert spaces, ...
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Unitary transformation in quantum mechanics

What two arbitrary states in the same Hilbert space can be connected through an unitary transformation? And how to construct the unitary transformation? Whether is there a general answers for these ...
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Derive Hamilton's Principle of Stationary Action Only from Unitarity in Quantum Mechanics?

In this Question I want to give a derivation of Hamiltons Principle of Stationary action, and my question to the community would be, whether my argument is flawed. The System I want to look at is (...
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Weinberg's “Derivation” of Lie algebra commutation relations

I have a question regarding the evolution of Lie algebra conditions in Weinberg's The Quantum Theory of Fields vol. 1: Foundations, chapter 2. I will reproduce the text here and state my two ...
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Is Hawking radiation truly random or we just don't know that it is?

First off I am not a black hole scientist and my education is limited to special relativity so please treat this question appropriately. The way I understand the BH information paradox is that ...
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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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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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How can I prove that the wave function remains normalized as time goes?

Exploiting Schrödinger equation and its conjugate we can show that $$ \dot{\Psi} = \frac{i \hbar}{2m} \nabla^2 \Psi - \frac{i}{\hbar} U \Psi $$ $$ \dot{\Psi}^* = -\frac{i \hbar}{2m} \nabla^2 \Psi^* + \...
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Do basis transformations of fields in QFT need to be unitary?

I know that it is possible to diagonalise kinetic terms in a QFT Lagrangian using generally non-unitary matrices to transform fields. For example https://physics.stackexchange.com/a/377299/91934. ...
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Taylor series for unitary operator in Weinberg

On page 54 of Weinberg's QFT I, he says that an element $T(\theta)$ of a connected Lie group can be represented by a unitary operator $U(T(\theta))$ acting on the physical Hilbert space. Near the ...
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Stone’s Theorem and Time Ordered Exponentials

The time evolution operator of quantum mechanics seems (at least to me) to form a strongly continuous, one parameter group of unitary operators. Hence, by Stone’s theorem, we should have that $U(t) = \...
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Does anything guarantee that a field theory will have a lower bound on energy, so that a vacuum exists?

If a system of particles is bound, then it has negative energy relative to the same system disassembled into its separated parts. In the nonrelativistic limit, this negative energy is small compared ...
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Commutator under unitary transformation

How can I prove that the commutators are invariant under unitary transformations? I'm studying quantum mechanics, so (maybe) my professor is talking about the commutator of hermitian operators.
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Canonical Transformations in Quantum Field Theory

In his lecture notes on canonical transformations in quantum field theory, Massimo Blasone defines the boson translation transformation to be \begin{equation} a_k \rightarrow a_k(\theta) = a_k + \...
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Factorisation of tree level amplitudes from unitarity

Is there a simple argument to explain why tree level amplitudes must factorize on their pole into products of lower point tree level amplitudes, not by ispection of Feynman diagrams but as a ...
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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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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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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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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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Unitary evolution from a mixed state to a pure state

Why is it not possible to have an unitary evolution from a mixed state to a pure state ?
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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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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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Why is information indestructible?

I really can't understand what Leonard Susskind means when he says in the video Leonard Susskind on The World As Hologram that information is indestructible. Is that information that is lost, through ...
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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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