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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79
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8answers
21k views

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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2answers
6k views

Why is time evolution unitary?

Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular ...
145
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2answers
21k views

Why do we not have spin greater than 2?

It is commonly asserted that no consistent, interacting quantum field theory can be constructed with fields that have spin greater than 2 (possibly with some allusion to renormalization). I've also ...
17
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5answers
1k views

Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
21
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3answers
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Why is the Yang-Mills gauge group assumed compact and semi-simple?

What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories? I'd think that these hypotheses lead to physically "...
9
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1answer
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Clarifications needed on Gauge Fixing and Ghosts [closed]

The first time some kind of gauge fixing appears is during the Gupta-Bleuler procedure, which is used to be able to quantize the photon field: The basic gauge invariant Lagrangian leads to $\Pi_0=0$ ...
6
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2answers
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Why are scattering matrices unitary?

In Griffith's QM book, he introduces scattering matrices as an end-of-the-chapter Problem 2.52. For a Dirac-Delta potential $V(x) = \alpha \delta (x - x_0)$, I've derived the scattering matrix and ...
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2answers
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Why does the action have to be hermitian?

The hermiticity of operators of observables, e.g. the Hamiltonian, in QM is usually justified by saying that the eigenvalues must be real valued. I know that the Lagrangian is just a Legendre ...
7
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3answers
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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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2answers
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Radial Schrodinger equation with inverse power law potential

Recently I read a paper about solving radial Schrodinger equation with inverse power law potential. Consider the radial Schrodinger equation(simply set $\mu=\hbar=1$): $$\left(-\frac{1}{2}\Delta+V(\...
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4answers
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Does Wick rotation work for quantum gravity?

Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.
7
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1answer
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Is the Lagrangian density in field theory real?

As the Lagrangian in classical mechanics corresponds to energy, it must be real. But is that the case in quantum field theory? I mean, it should still correspond to some sort of energy, but what about ...
5
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1answer
2k views

Holstein-Primakoff and Dyson-Maleev representation

In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
8
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2answers
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Sign in front of QFT kinetic terms

I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} \...
6
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1answer
901 views

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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How do derivative couplings affect canonical quantization?

Consider a Lagrangian for a scalar field $\phi$ with an interaction term $$\mathcal{L}_{int} = (\partial^2 \phi)^2 \phi.$$ Here I'm suppressing all indices for brevity. Now, this is just a three-...
5
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1answer
452 views

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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What exactly does No cloning mean, in the context of Quantum Computing?

I am trying to get an intuitive idea of how the No-Cloning theorem affects Quantum computation. My understanding is that given a qubit $Q$ in superposition $Q_0 \left| 0 \right> + Q_1 \left| 1 \...
3
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3answers
960 views

Shankar's Active/Passive Change of Basis

I'm working my way through Shankar's Quantum Mechanics (7th printing, and I'm doing it alone, so I apologize if I have core concepts completely wrong). He has a section on Active and Passive ...
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2answers
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Why do Faddeev-Popov ghosts decouple in BRST?

Why do Faddeev-Popov ghosts decouple in BRST? What is the physical reason behind it? Not just the mathematical reason. If BRST quantization is specifically engineered to make the ghosts decouple, how ...
10
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1answer
855 views

Basic question about the S-Matrix, Unitarity and Effective Field Theory

Consider scattering some particles in a state collectively denoted by $i$ to a final state denote by $f$. The scattering amplitude, S-matrix is then defined by: $S_{fi}\equiv \langle f|e^{-iHt}|i\...
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3answers
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Quantum mechanics - how can the energy be complex?

In section 134 of Vol. 3 (Quantum Mechanics), Landau and Lifshitz make the energy complex in order to describe a particle that can decay: $$ E = E_0 - \frac{1}{2}i \Gamma. $$ The propagator $U(t) = \...
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1answer
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Unitary quantum field theory

What do physicists mean when they refer to a quantum field theory being unitary? Does this mean that all the symmetry groups of the theory act via unitary representations? I would appreciate if one ...
13
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2answers
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Why does unitarity require the Higgs to exist?

A standard argument that the Higgs boson must exist is that without it, amplitudes in the Standard Model at the TeV scale violate unitarity. This is explained in section 21.2 of Peskin and Schroeder ...
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3answers
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Relation between unitarity and conservation of probability

In a seminar, I heard that the unitary aspect of representations was important physically, because in quantum mechanics unitarity is closely tied to the conservation of probability. Could someone ...
8
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2answers
350 views

How can an inverted anharmonic potential $V(x)=-x^4$ have discrete bound states?

I've been watching the lectures on mathematical physics by Carl Bender on youtube where he uses the non-Hermitian Hamiltonian methods to prove that the inverted anharmonic potential $V(x)=-x^4$ has a ...
5
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0answers
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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 ...
11
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1answer
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Why do quantum gates have to be reversible?

One possible reason I have come up with is that we are modeling quantum gates by unitary matrices. And since unitary operations are reversible we have to be able reverse the operation in the physical ...
76
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10answers
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Is there a symmetry associated to the conservation of information?

Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory. We may wonder if there is an underlying ...
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3answers
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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 ...
35
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1answer
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Is general relativity holonomic?

Is it meaningful to ask whether general relativity is holonomic or nonholonomic, and if so, which is it? If not, then does the question become meaningful if, rather than the full dynamics of the ...
13
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1answer
954 views

QED and anomaly

I've just started to learn anomalies in quantum field theories. I have a question. How to show that QED is free from vector current anomaly and what would happen if it were not? In other words, how ...
12
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2answers
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Irreducible Representations Of Lorentz Group

In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as $P^{...
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1answer
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Is it possible to derive Schrodinger equation in this way?

Let's have wave-function $\lvert \psi \rangle$. The full probability is equal to one: $$\langle \Psi\lvert\Psi \rangle = 1.\tag{1}$$ We need to introduce time evolution of $\Psi $; we know it in ...
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3answers
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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 ...
5
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1answer
362 views

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 ...
11
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2answers
788 views

Scale invariance plus unitarity implies conformal invariance?

What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?
13
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1answer
761 views

Probability conservation in WKB tunneling

Suppose we have quantum mechanical plane waves of energy $E$ incident upon a one-dimensional potential barrier $V(x)$ with sloping sides. One can compare the WKB solutions in the three relevant ...
7
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1answer
615 views

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?
6
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1answer
608 views

Positivity of residues and unitarity in scattering amplitudes

I am reading "Superstring Theory" by Green, Schwarz, Witten. In the introduction, about the Veneziano amplitude (below eq. 1.1.16/17), they say that The residues of poles must be positive in a ...
4
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0answers
416 views

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 $...
6
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1answer
639 views

Why is $R^2$ gravity not unitary?

I have often heard that $R^2$ gravity (as studied by Stelle) is renormalisable but not unitary. My question is: what is it that causes the theory to suffer from problems with unitarity? My naive ...
4
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3answers
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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) ...
10
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2answers
807 views

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 ...
7
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1answer
360 views

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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1answer
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Why does the state space contain states with negative norm and what would be an example?

My lecture script of Quantum Field Theory states that " the state space contains states with negative norm ". Why does it have to be like this and what would be an example fo such a state?
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1answer
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Prove that this operator is unitary

$$\hat{O}\equiv(1/\sqrt{2\pi})\int e^{-iNz}dz$$ $$\hat{O}^\dagger\equiv(1/\sqrt{2\pi})\int e^{iN'x}dx$$ We have the operator $\hat{O}$ and its Hermitian adjoint $\hat{O}^\dagger$, in the one ...
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0answers
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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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0answers
410 views

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-...
4
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1answer
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Embedding of particles into fields

For the classification of particles (Wigner 1939), we look for unitary representations of the Poincaré/Lorentz group. There are only infinite-dimensional (non-trivial) unitary representations! To ...