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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Tensor product of universe Hilbert space and black hole Hilbert space equals?

I'm a newbie struggling to parse concepts important in black hole, holographic principle and related issues. My query comes from Thomas Thiemann’s Modern Canonical Quantum Gen. Relativity, which ...
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Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
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Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
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Interactions terms among vector fields in Lagrangian

In the QED Lagrangian we have terms that look like $\partial_{\nu} A_{\mu}$ and $J_{\nu} A_{\mu}$. We cannot have $A_{\mu} A^{\mu}$ as its coefficient would give a mass term, and for the photon we ...
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The necessity of ground state in QFT

Why we always want to have a ground state in every physical theory? For example, when we try to quantize Dirac Hamiltonian and encounter a Hamiltonian without a ground state, we take a step back and ...
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Show $S$-operator is unitary

In an exercise, we are supposed to show that the scattering matrix on the right of $$S_1(E)= \begin{pmatrix}t_1 & r_1' \\ r_1& t_1'\end{pmatrix}\delta(E_f-E_i)$$ is unitary. We are explicitly ...
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Why are symmetry transformations connected to the identity necessarily represented by linear unitary operators?

I'm just trying to wrap my head around the following paragraph (taken from "The Quantum Theory of Fields", Weinberg, Vol. 1, Ch.2): There is always a trivial symmetry transformation, $\mathscr{R}\...
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174 views

Why is time evolution unitary in $PT$-symmetry?

I have a question on the time evolution for a $PT$-symmetric Hamiltonian. So far I have only read that time evolution was unitary because $H$ commutes with $PT$ and the newly constructed operator $C$ ...
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Connecting the Schrodinger equation, unitarity and norm-preserving of states in time evolution

This may be really basic but I'm having trouble connecting the following issues: 1) The 2-norm for state vectors is preserved during time evolution 2) The Hamiltonian is a Hermitian operator. With ...
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Unitarity of Quantum Mechanical Systems

I was reading this lecture notes "Black holes from A to Z" by Andrew Strominger. In the first chapter Introduction the following statement is made: "If we know the present, there are laws that ...
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Why do unitary transformations on ensembles of states result in the same density matrix?

This is from Nielsen and Chuang. If we have an ensemble of pure states that obey the following relationship for all $i, j$ $\vert \psi_i \rangle = \sum_{j} u_{ij}\vert \phi_j \rangle$ where $u_{ij}$...
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What does it mean for the Hamiltonian to not be bounded from below?

According to David Tong. In quantum field theory if you quantize the Dirac field using commutation relations instead of anti-commutation relations you end up an unbounded Hamiltonian from below, page ...
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Quantizing the Dirac field using commutation relations leads to an unbounded Hamiltonian?

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 ...
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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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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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Why is the $\varepsilon^2$ term in an infinitesimal transformation equal to zero?

Given the unitary operator $U=1+i\varepsilon F$ (where $\varepsilon$ is an infinitesimal scalar), in order to prove that $F$ is Hermitian: $$\begin{align} UU^{\dagger} &= 1 \\ (1+i\varepsilon F) (...
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Time ordering operator and derivative with respect to time

In the book Quantum field theory and the standard model from Schwartz, it is written on page 87 some results using time ordering operator. We have the following operator: $$ U(t,t_0)=T \exp\biggl(-\...
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Symmetry transformations on a quantum system; Definitions

We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a ...
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Why is quantum mechanics reversible?

"Quantum mechanics is reversible" this statement is everywhere, some even said it's just an observed fact about the universe. I can't find a layman explanation or example why is it reversible?
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Unitary time condition

I have a confusion with regards to the principle of QM that states that time evolution must be unitary. In particular, given that states transform through time as $|\Psi(t)\rangle = U(t)|\Psi(0)\...
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On finite-dimensional unitary representations of non-compact Lie groups

In this thread Lorentz transformations for spinors, V. Moretti made a claim as follows: "it is possible to prove that no non-trivial finite-dimensional unitary representation exists for a non-compact ...
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Lorentz transformations for spinors

The lorentz transform for spinors is not unitary, that is $S(\Lambda)^{\dagger}\neq S(\Lambda)^{-1}$. I understand that this is because it is impossible to choose a representation of the Clifford ...
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Unitary rotation of spin states

Consider the $2j+1$-dimensional Hilbert space spanned by the spin states $$\left|j,-j\right>,\left|j,-j+1\right>,\ldots,\left|j,j-1\right>,\left|j,j\right>$$ where $\hbar m$ and $\hbar^...
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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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In QFT, can Field Operators at different points in Space-time always be expressed as unitary Transformation of each other?

Given an operator valued field $\Phi(x)$, and two points in spacetime, $x$ and $y$, can I always write down something like: $$ \Phi(y) = U_{x,y}^{-1} \Phi(x) U_{x,y} $$ With $U_{x,y}$ being an ...
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Do I have to transform all Observables (all Operators) simultaneously?

Let's say I have a quantum system with 2 observables $\hat{O}_1$ and $\hat{O}_2$. Those are supposed to be "functions"of $\hat{X}$ and $\hat{P}$. Let's say I want to Transform $\hat{O}_1$ using an ...
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Semiboundedness of Schrödinger operators

Is the Schrödinger operator $$H~=~-\Delta+V$$ bounded from below? For example, I would like to analyze the case where $V\in L^{2}_\mathrm{loc}(\mathbb{R}^{n})$ is a locally square integrable function, ...
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Is the coherence of a quantum state a relative concept?

It is often said that e.g., in describing the collapse of states in Quantum Mechanics (QM), speaking or analyzing in terms of information provides a more solid footing compared to focusing on changes ...
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Local unitary transformation of pure two-qubits system

In the following paper https://arxiv.org/pdf/0707.1780.pdf, at the beginning of P.2, the author is saying that: For any states of pure two-qubit systems, $\left| \psi \right\rangle = \alpha_{00} \...
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Understanding how the quartic Higgs coupling turns negative at high energies

How is the conclusion that the Higgs quartic coupling becomes negative at high energies ($\sim 10^{10}$) GeV inferred? I'm looking for a reference that has the necessary idea and calculations to ...
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Why is time-evolution operator unitary?

When we shift the system's time from $t=0$ to $t = t$, we can define the following operator $\hat{U}$. $$\hat{U} = e^{- i \hat{H} t / \hbar} \, .\tag{1}$$ So many (as far as I read, almost all of) ...
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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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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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Can we make sense of a Hamiltonian $a^\dagger a^\dagger + a a$?

If I have a Hamiltonian given by $$ H = a^\dagger a^\dagger + a a$$ where, $[a,a^\dagger] = 1$, Can I make sense of it, by generalizing the notion of vacuum? If not what sort of troubles I would run ...
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How can I generate a random walk on $U(n)$?

I asked this question here on math.SE: https://math.stackexchange.com/q/2250448/78169 I'm asking in the physics forum in order to get a different perspective, and also as I suspect it's likely that ...
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How the useful, unitary “Folding Transform” is applied to a Hamiltonian

Summary Given a unitary transformation $U_F(t)=\sum_n e^{in\omega t}\sum_{\lambda\in n}\left| \lambda, n\right\rangle \left\langle \lambda,n \right|$ applied to a Hamiltonian $H_0$ (with $H_0 \left|\...
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Why do negative norm states break unitarity?

I often hear my teachers say that the negative norm states break unitarity. And I can also read this elsewhere, such as at this place In this gauge the relation between unitarity and gauge ...
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Non-hermitian hamiltonians and symmetry properties of $\partial_t$ [duplicate]

From this question another one came to my mind. Consider the Hilbert space $\mathcal{H}$ of function square integrables $\psi:\mathbb{R}^n\rightarrow\mathbb{C}$ with the usual inner product: $$ \...
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Schrödinger equation and non-Hermitian Hamiltonians

Is the Schrödinger equation still valid if we use a non-Hermitian Hamiltonian with it? By this I mean does: $$\hat{H}\psi(t) = i\hbar\frac{\partial}{\partial t}\psi(t)$$ if $\hat{H}$ is not ...
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How can information ever get lost at the event horizon of a black hole?

In the drawing, A and B are two entangled particles in Kruskal coordinates, A is falling into the black hole, B is remaining outside. The lines going through the center are the time coordinates of ...
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Does the twin paradox ensure the quantum unitarity near a black hole?

A quantum system of two entangled particles is located near a black hole, particle A is receding from and particle B is moving towards the event horizon. When particle B is crossing the event horizon,...
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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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Allowed Field Re-definitions in QFT

I am trying to understand which field redefinitions are allowed in a QFT. The textbooks I have read appear to treat this topic flippantly. I assume that one cannot arbitrarily manipulate the ...
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Dirac equation, why not unitary, why not single-particle formalism?

I am reading the first chapter of Akhiezer, Berestetskii QED (1981). They state that Dirac was wrong to assume that the evolution of the wave function is described by $\psi(t) = e^{-iHt} \psi(t_0)$ ...
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Why is time-evolution unitary (the sequel)?

One foundational postulate of QM is that a closed physical system at one instant of time, say $t$, is completely described by a wavefunction $\psi \in S^1\subset H$ (where $H$ is a Hilbert space and $...
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The Ward identity, the Lorentz invariance

Outline - heuristic derivation of the Ward identity from the requirement of the Lorentz invariance Suppose we have the free quantized gauge theory (with quanta called photons) with the Hilbert space ...
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Cutkosky rule for the triangle diagram

Outline - the anomalous vacuum polarization correction Suppose the abelian anomalous gauge theory (with axial gauge field $A$, vector gauge field $V$ and single massless fermion $\psi$): $$ \tag 1 L =...
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Optical theorem for the given diagram

Assume we have the abelian gauge theory with single fermion. Suppose the following diagram: Here the initial "photon" $\gamma$ is in the same state as the final one, so this is the diagram of self-...
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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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Gauge invariance and the unitarity

I want to discuss the relation between the unitarity and the gauge invariance. Suppose we have for simplicity an abelian gauge theory (say, EM theory). We want to quantize it in terms of 4-potential $...