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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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Uniform dynamics in quantum mechanics

I've found in the book "Quantum Processes System & Information" of Benjamin Schumacher the following definition of "uniform dynamic": it often happens that the basic dynamical ...
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Relationship between unitaries generated by a Hamiltonian and its negative sign

Consider two unitary operations $U_1$ and $U_2$ defined by: $\partial_t U_1 = -iH_1U_1$ and $\partial_t U_2 = iH_1U_2$ Here, $U_1$ is generated by $H_1$ and $U_2$ is generated by $-H_1$, with the ...
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Fermi theory cutoff from unitarity bound

Tree-level cross sections for processes described by Fermi theory behave like $\sigma $ $\sim$ $G_{F}^2 \cdot s$, where $G_{F}$ is the Fermi constant and $\sqrt s$ is the energy entering in the ...
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Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?

I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12: $$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
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Finiteness of the Kac table for minimal model

I am currently reading CFT from Di Francesco. I am stuck at not understanding why the kac table for minimal models $(p, q)$ where $p$ and $q$ are co-prime, only has fields with conformal dimension $h_{...
Suriyah R K's user avatar
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Is it possible for a unitary transformation to align 2 spatially distinct wave packages into 1?

This is mainly meant as a more concise and more general formulation of the problems and realizations occurring to me while thinking about the apparatus I described in this question. The main problem ...
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Unitarity and renormalizability in $R_\xi$ and 't Hooft gauge

Consider the massive propagator with gauge fixing $\frac{1}{2a} (\partial A)^2$ $$ \Delta_{\mu\nu}=-i\left[\frac{g_{\mu\nu}}{k^2-m^2}-\frac{k_\mu k_\nu}{m^2}\left(\frac{1}{k^2-m^2}-\frac{1}{k^2-am^2}\...
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Motivation behind reflection positivity

I have taken a look at this physicsSE question, Wikipedia, and this paper by Jaffe which go over reflection positivity. While they all nicely explain the definition behind reflection positivity and ...
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Translation operators and positive-semidefinite condition

Good day. I have an operator $\exp{(-i\mu\hat{q})}\exp{(-i\nu\hat{p})}$, where $\hat{q}$, $\hat{p}$ are the position and momentum operators, respectively. The parameters $\mu,\nu$ are some real ...
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Inverting an integral relation involving a set of continuous modes

Consider a continuous system of bosonic modes $a(x)$, and the transformation $$U a^\dagger(x) U^\dagger = \int A(x,y)a^\dagger(y) \ dy. $$ My goal is to find $U^\dagger a^\dagger(x) U$ in terms of a ...
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Does the Hamiltonian always commute with the Time Evolution Operator?

The time evolution operator $U(t, t_0)$ is given as the solution of the equation $$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$ whether or not the system is conservative. When the ...
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Is stochasticity totally incompatible with current theories of conservation of information in quantum physics?

Most physicists accept the unitarity principle of the universe, according to which the state of a system at any given time must determine its state at any other time (information is never lost, ...
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Generator of time shift when the Hamiltonian is time dependent

Let's consider the unitary group $\hat{S_{\tau}^†}$ such that :$$\hat{S^†_{\tau}}|\psi(t)\rangle=|\psi(t-\tau)\rangle$$ Since we know that: $$\hat{U}(t,t_0)|\psi(t_0)\rangle=|\psi(t)\rangle$$ Where ${...
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Unitary evolution of the displacement operator in terms of positions and momenta

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
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How does path integral quantization ensure unitarity?

Unitarity can be verified post hoc by examining the optical theorem. In the context of path integral quantization where formal derivation starting from canonical quantization is unavailable, is it ...
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Unitary Representation of $\text{SO}(3)$ in Position Representation

Let $R\in\text{SO}(3)$ be an arbitrary rotation, and let $U_R$ be the unitary representation of $R$ on some Hilbert space $\mathcal H$. To me, the defining property of $U_R$ is how it conjugates the ...
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Coherent creation operator: unitary or not?

In Quantum Mechanics, for coherent states $|z\rangle$ it can be prooved that if $|0\rangle$ is the vacuum state for an harmonic oscillator, therefore: \begin{equation} |z\rangle=e^{za^{\dagger}-z^*a}|...
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On the Wigner symmetry representation theorem

Wigner symmetry representation theorem tells that if $\mathcal{S}:\mathbb{P}\mathcal{H}\to \mathbb{P}\mathcal{H}$ is a symmetry, then $\mathcal{S}[\Psi]=[\hat{U}\Psi]$ where $\hat{U}:\mathcal{H}\to \...
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Connection of a concrete Hamiltonian to the generator of time-translations

In a Quantum-mechanics lecture I am hearing we defined the Hamiltonian of a quantum system (a system with an observer) as the generator of the time translation-operation of the system under ...
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Discontinuity of the scattering amplitude and optical theorem

The generalized optical theorem is given by: \begin{equation}\label{eq:optical_theorem} M(i\to f) - M^*(f\to i) = i \sum_X \int d\Pi_X (2\pi)^4 \delta^4(p_i-p_X)M(i\to X)M^*(f\to X).\tag{Box 24.1} ...
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Adjoint and index notation in Weyl field context

In the answer to a question I previously asked, the following manipulation was done but I don’t understand it$.$ $$ (U_{jm}\psi_m)^\dagger=\psi_m^\dagger U_{mj}^\dagger $$ aside from the context from ...
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Unitarity in the 't Hooft limit

Consider a quantum gauge theory with a holographic dual at infinite $N$ and 't Hooft coupling, in which the gauge theory is described by classical (super)gravity. If I initialize the system in a pure ...
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Branch cut of a one-loop bubble diagram after cutting a single propagator

I am trying to understand Cutkosky cutting rules and generalized unitarity. Consider the article https://arxiv.org/abs/0808.1446 by Arkani-Hamed, Cachazo & Kaplan. In chapter 5.1 equation 133, the ...
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Strange result with unitary transformation

If the unitary operation $\hat{U}$ turns the bases $|a\rangle$ to $|b\rangle$ as $\hat{U}|a\rangle =|b\rangle$ then we have: $$\hat{U}=\sum_{aa'}|a'\rangle\langle a'|\hat{U}|a\rangle\langle a|=\sum_{...
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Condition on unitary operator for real eigenstates of Hamiltonian

I'm working with the discrete-time quantum walk in which the evolution is described by the unitary operator - $$U = S(C\otimes I)$$ where $C$ is the coin operator (acts on spin degree of freedom of ...
Young Kindaichi's user avatar
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Beam splitter with complex parameter

A non-polarizing beam splitter can usually be described by a unitary operator such as $U=e^{i\theta(a^\dagger b+b^\dagger a)}$ given a parameter $\theta\in \mathbb R$ and a pair of independent modes $...
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Composing beam splitters

Let $a, b$ and $c$ be independent modes in a system $S$ and in environments $E_1$, $E_2$ respectively. Suppose $a$ goes through a beam-splitter characterized by a parameter $\theta$ coupling it to ...
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Checking $U^{\dagger} U=I$ for Lorentz boost

The boost is given by $|p\rangle \rightarrow |\Lambda p\rangle$. I can understand that this is unitary because the inner product : $$\int dp \frac{1}{2\omega _p} f(p) g^*(p) ....(1)$$ is invariant. I'...
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Confusion on Hamiltonian unbounded from below and Ostrogradsky Instability

This might be a silly question but I failed to get it. In Ostrogradsky Instability, we deduced that Lagrangian of higher-order derivatives leads to Hamiltonian linear to canonical momenta, and thus, ...
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What is a general expression for a symmetric $N$-mode beam splitter?

BACKGROUND A symmetric beam splitter can be represented as \begin{equation} \hat{B}^{(2)} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \end{equation} and according ...
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Wigner's theorem for 2-state system

I am trying to see how Wigner's theorem can be proven in a 2-state system. Let, $$ |\psi\rangle= a |0\rangle+b|1\rangle, \quad|\phi\rangle= c |0\rangle+d|1\rangle $$ where all coefficients are complex ...
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Arbitrary $2\times 2$ unitary matrix using waveplates

I'm studying about the quantum computation using quantum optics and I wonder about how to implement arbitrary $2\times 2$ unitary operation using waveplates such as half wave plates and quarter wave ...
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Anticommutation relations for Dirac field at non-equal times

I'm reading this paper by Alfredo Iorio and I have a doubt concerning the anticommutation relations he uses for the Dirac field. Around eq. (2.25), he wants to find the unitary operator $U$ that ...
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What guarantees a photonic quantum gate to be unitary?

So in my course of quantum computation i came across this question that "What guarantees a quantum gate to be unitary?" i was specially curious about photonic quantum gate. At first i ...
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Writing a given unitary in the same basis as the Hamiltonian (Operator Representation and Confusion)

I have a simple question concerning how to write the representation of operators, such as unitaries, using a specific order for the basis elements. Let me give you an example. Consider a tripartite ...
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Normalization of the harmonic oscillator propagator

The propagator of a quantum system is defined by $$\mathcal{K}(t,x;\,t_{0},x_{0})\,\equiv\,\left\langle x\right|\hat{U}(t,\,t_{0})\left|x_{0}\right\rangle.$$ In this notation, the unitarity demands ...
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Why do we expect unitarity to be preserved in the black hole information paradox?

Consider the following way of describing the black hole information paradox: Suppose we start with a pure quantum state and a black hole of mass $M$. Now we throw the pure state into the black hole ...
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Can you apply non-unitary operators to a qubit?

I am wondering if it is possible to apply continuous, invertible transformations to a qubit which are not linear, i.e. not elements of $U(N)$ where $N=2^n$ where we have $n$ qubits. Consider $n=1$. ...
Jackson Walters's user avatar
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Measurement problem and precise mathematical calculation

The infamous measurement problem is a problem in the foundations of quantum mechanics: different people have different views how to understand this problem: some people even deny that there is any ...
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Why must the propagator exponent be imaginary?

In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
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Basis change in the context of the Schroedinger and Heisenberg picture

I am trying to understand the following statement: One sometimes summarizes this situation with the slogan: In the Schr¨odinger picture the basis vectors (provided by the observables) are fixed, while ...
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How is the Wigner little group representation of Poincaré group Unitary?

From Weinberg's QFT Vol.1, eq(2.5.11): $$U(\Lambda)\Psi_{p,\sigma}=({N(p)\over N(\Lambda p)})\sum_{\sigma'}D_{\sigma'\sigma}(W(\Lambda,p))\Psi_{\Lambda p ,\sigma '}.\tag{2.5.11}$$ However, this is not ...
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Proof that different representations of gamma matrices are connected by a unitary transformation

The different representations of gamma matrices are related by a similarity transformation \begin{equation} \gamma^{\mu'}=S\gamma^{\mu}S^{-1} \end{equation} for some non-singular $S$. I have to prove ...
Anindita Sarkar's user avatar
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Is there a physical reason why the time evolution in quantum mechanics is given by $e^{-itH}$? [duplicate]

Let $H$ be the Hamiltonian operator. Since $H$ is self-adjoint, by Stone's theorem there is a strongly continuous one-parameter unitary group $U(t)$ such that $U(t) = e^{-itH}$. Mathematically this ...
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Variation of the Ricci Tensor

In my own research into the use of Clifford Algebras with the Standard Model, gravity appears, but in calculating the Lagrangian of the theory I get: $$-\frac{3}{4}\sqrt{-g}R^{\alpha\beta}R_{\alpha\...
Jason's user avatar
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The unitarity of the $\delta(x)$ potential

One of the common potentials to solve in quantum mechanics is when $$H=\frac{p^{2}}{2m}+\delta(x).$$ Is this Hamiltonian considered to produce unitary evolution? In particular, I'm not sure what is ...
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CKM Matrix Magnitude Question - Top to Bottom Greater than One?

In the CKM Matrix article on Wikipedia, the "standard" parameters for each matrix element are written with combinations of sines, cosines, and a complex exponential of 4 angle values. Using ...
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Kinetic Terms for Interacting Massive Vector Field

The kinetic term in the standard Lagrangian for a vector field, whether massive or not, is $$-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}=-\frac{1}{2}(\partial_\alpha V_\beta \partial^\alpha V^\beta - \...
Ahron Maline's user avatar
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Is unitary time evolution the same as obeying the Schrödinger equation?

In this question, the answer says that unitary time evolution means that probability is conserved. Is this the same as saying that a system obeys the Schrödinger equation?
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Is linearity sufficient to guarantee probabilities are invariant under change of reference frame?

In his Symmetry Principles in Quantum Physics, Fonda seems to write (in a footnote at the bottom of page 13) that if a bijection $T$ on a vector space is linear or antilinear, then for any $\phi,\psi$ ...
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