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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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 $n=1$. ...
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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 ...
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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:
$$-\frac{3}{4}\sqrt{-g}R^{\alpha\beta}R_{\alpha\...
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The unitarity of the $\delta(x)$ potential
One of the common potentials to solve in quantum mechanics is when $$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 - \...
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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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Integral of the square of the spectral density in a quantum field theory
The quantity of interest is
$$
\int_0^\infty dE \, \rho(E)^2
$$
where $\rho(E)$ is the spectral density in a Quantum Field Theory. I am wondering whether
it has any physical meaning, and
it can be ...
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Is unitarity equivalent to linearity plus conservation of the norm?
Unitarity is the condition that the inner product in the Hilbert space is preserved. But if you suppose that the norm of any state is already preserved, then does unitarity follow from linearity? ...
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Are the generators of the $SU(2)_L\times SU(2)_R$ group unitary?
In the chiral model, the quark field $q=\begin{pmatrix} u \\ d \end {pmatrix}$ transforms like $q\rightarrow\exp({i(\theta_a\tau^a+\gamma_5\beta_a\tau^a)})\;q$. Now, I understand that the ...
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The use of unitary operators in translating between reference frames in quantum mechanics
In Chapter 2.6 of his Lectures on groups and vector spaces for physicists, Isham describes how different observers (different reference frames) must describe the same quantum system.
Consider two ...
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Can a non-unitary process be made unitary using a transformation or by expanding the phase space?
Suppose I have a matrix differential equation:
$$ \frac{d\mathbf{x}}{dt} = A\mathbf{x}$$
The solution to this is
$$\mathbf{x}(t) = e^{At}\mathbf{x}(0)$$
If $A^{\dagger}=-A$, then the time evolution ...
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Uniform losses commute with linear optics: how does it work?
In many papers about quantum optics and interferometry, it's assumed or said that "it's well known" that linear optics commutes with uniform losses. In particular if we have a beam splitter ...
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Dimensionality of unitary representation of Lorentz group
I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 137, he talks about the non-unitarity of the finite-dimensional representation of the ...
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States created by local unitaries in QFT
In quantum field theory, consider acting on the vacuum with a local unitary operator that belongs to the local operator algebra associated with a region. In such a way, can we obtain a state that is ...
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Unitary representations of a Lorentz transformation
In QFT we have an action of the restricted Lorentz group which is implemented via a unitary transformation. In other words, if $\Lambda\in SO(1,3)^\uparrow$, then the corresponding unitary operator is ...
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Derivations about Time-Dependent Perturbation Theory (Griffiths)
I've been reading the Time-Dependent Perturbation Theory in Griffiths' book. There are many places that puzzle me a lot.
The system is two-level($\psi_a,\psi_b,E_a<E_b$). Suppose the particle ...
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Is quantum gravity compatible with unitary evolution?
I am thinking that they aren't strictly compatible. I have the following logical argument for this:
The unitary evolution postulate says that the state of a system is given by a time-depending state ...
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Is $U(n)\simeq SU(n)\times U(1)$ [duplicate]
As the title suggests, my question is does $U(n)\simeq SU(n)\times U(1)$ for general $n$?
If so, how does one prove it?
If not, is at least $U(n)\supset SU(n)\times U(1)$? What is the group $\frac{U(n)...
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Why is a symmetry transformation defined to preserve $|\langle\phi|\psi\rangle|^2$?
In the proof of Wigner's theorem, the crucial role is played by the quantity $|\langle\phi|\psi\rangle|^2$ where $|\psi\rangle$ and $|\phi\rangle$ represent two distinct physical states (or more ...
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How causality and unitarity are ensured given a non-linear electromagnetic Lagrangian?
I am reading these notes on non-linear electrodynamics (NED). On page 8, below equation (5.1) the author states that the modified electromagnetism parameter $\gamma$ should be non-negative in order to ...
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Self-adjointness, time evolution operator and the role of domains
Introduction
Let us consider a Hamiltonian $\hat{H}$ for a certain system and suppose that I would like to know if it is possible to define it (i.e. its domain) in such a way that it results self-...
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Corners in the worldsheet and Ricci scalar term in the Polyakov action
I have a question about a passage in Polchinski's textbook [1], regarding the topological term in the Polyakov action.
In the Polyakov action for a closed manifold, we can add a term proportional to ...
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How does $e^{-i\vec a \cdot \vec p}$ translate a quantum system? [duplicate]
I'm reading Conceptual Framework of QFT by Anthony Duncan, and at the bottom of page 77 he says
The translation of a physical system by displacement $\vec a$ is of course effected by the unitary ...
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Non-unitary magnetic translation operator for Landau levels
In chapter 9 of their book [1], Altland and Simons consider the Landau Hamiltonian in the symmetric gauge,
\begin{equation}
H=\frac{1}{2m^*}\left[\left(-\mathrm{i}\partial_1-\frac{x_2}{2l_0^2}\right)^...
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What does sandwiching with an unitary operator and its inverse imply?
I am following the book "An introduction to quantum field theory" by Peskin and Schroeder. In the section 'Discrete symmetries of the Dirac theory', it is written,
$P a^s _p P^{-1} = \eta_a ...
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Is unitarity intimately "connected" to symmetry?
I have a couple of questions about unitarity and symmetries:
Is unitarity connected to all fundamental symmetries? Is it linked to symmetries like CPT, Lorentz, Poincaré, diffeomorphism, ...
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How to transform a Lindblad operator basis?
I'm trying to understand how to perform a unitary transformation on a set of traceless orthonormal Lindblad operators, following chapter 3.2.2 of The Theory of Open Quantum Systems by Breuer and ...
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What unitary operator achieves Bogoliubov transformation?
Let $a_1, \ldots, a_N$ be boson operators, $[a_i, a_j^\dagger ] = \delta_{ij}$. One often considers Bogoliubov transformation $a_i \to \sum_j (A_{ij} a_j + B_{ij} a_j^\dagger)$, where the matrices $A=(...
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Why Lorentz transf. representation in spin 1/2 particles Hilbert space is not a unitary operator? [duplicate]
Weinberg introduces the idea of Lorentz group representation describing how vectors in the Hilbert space of definite momentum states should change due to a L.T. It is understandable that to preserve ...
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Relationship between $S$ matrix elements and $T$ matrix elements
In hadron physics, we define the $S$ matrix as $S=I+iT$, where $T$ is transition matrix. But on the question: Finding relation between matrix $S$ and matrix $M$ for wave propagation, it gave an ...
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Projective representations of the Lorentz group can't occur in QFT! What's wrong with my argument?
In flat-space QFT, consider a spinor operator $\phi_i$, taken to lie at the origin. Given a Lorentz transformation $g$, we have
$$\tag{1} U(g)^\dagger \phi_i U(g) = D_{ij}(g)\phi_j$$
where $D_{ij}$ is ...
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Is there a way to convert a CNOT into a single Hilbert space unitary?
Imagine the following CNOT gate:
Knowing that the secondary system state is fixed to $|0\rangle \langle0$|, I have the feeling that it can be converted into the following single Hilbert space unitary:...
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Is Dyson Series a unitary operator?
I am currently study time dependent perturbation theory. If I understand correctly Dyson series help us to approximate time evolution of initial state.
However, I am confused about the case when we ...
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Scattering Amplitude & Unitarity
In Srednicki's Quantum Field Theory chapter 11, the probability of a $2 \to n$ scattering process is calculated to be
$$
P = \frac{|\left<f|i\right>|^2}{\left<f|f\right>\left<i|i\right&...
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Is the higgs metastability dependent on renormalization scheme?
It is said that, in the absence of physics beyond the Standard Model (SM), the higgs potential
$$V_h= -\mu^2 \Phi^\dagger \Phi +\lambda (\Phi^\dagger \Phi)^2$$
becomes metastable (is unbounded from ...
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Which operator measures "energy", when unitary transformations don't change matrix elements, but they change time evolution?
This question is a more general (and shorter) version of this previous question of mine. We know that from any quantum-mechanical description of a system, we can go to an equivalent description by ...
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Effect of applying Hermitian conjugate of inversion operator
I'm glad Victor Galitski's monolith is finally out in English version, but I was confused by the following problem:
$$
\text{Find the Hermitian conjugate of the inversion operator }\hat{\Pi}: \hat{\Pi}...
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Is this strenghthening of Wigner's theorem on quantum symmetries true?
For every nonzero vector $\phi$ in a Hilbert space $\mathcal{H}$, let us denote $[\phi] := \mathbb{C}\phi$ the ray associated to $\phi$.
Let $S$ be the set of rays. For all non-zero $\phi,\psi$, $<[...
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Is quantum cloning $|\psi\rangle|\psi_1\rangle|\psi_2\rangle|C\rangle\to e^{i\alpha}|\phi\rangle|\psi\rangle|\psi\rangle|C'\rangle$ prohibited?
I think the no-cloning theorem is too restrictive, as in,
$$|\psi\rangle |\phi\rangle\to e^{i\alpha}|\psi\rangle|\psi\rangle \tag{1}$$
does not allow for any arbitrariness in the final state.
Instead, ...
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Why does the Eigen-Energy-Shift for the AC-Stark-Shift, calculated in the "rotating frame", actually matter?
As is neatly described in this answer, the AC-Stark Shift is the observation that the energy-eigenvalues of stationary states in the "rotating frame" of a two-level-system behave as (Adam ...
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Faddeev-Popov ghost in the Standard-Model
When we quantize $SU(N)$ gauge theories using the path integral formalism, we must introduce Faddeev-Popov ghosts and will appears as scalar fermions coupled to our gauge bosons in the Lagrangian of ...