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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About Cutkosky cutting rule
I am working on a problem about Cutkosky’s cutting rule in Matt D. Schwartz’s Quantum Field Theory and the Standard Model. The problem asks us to show that the imaginary part of the amplitude is given ...
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Does Wigner's theorem only imply left inverse?
From wikipedia
https://en.wikipedia.org/wiki/Wigner%27s_theorem
For unitary case
$$\langle U \Psi, U \Phi \rangle = \langle \Psi, \Phi \rangle .\tag{1} $$
If I apply the definition of adjoint
https://...
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Vector transformations that preserve norms but changes inner products between different vectors
Unitary transformations conserve the inner product structure of a set of vectors, they only change the direction of the vectors, i.e. rotate them all in the same way. A unitary transformation $U$ can ...
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Confusion about Cutkosky rules
The standard Cutkosky rules go as follows:
Cut the appropriate propagators (replace them with $(-2\pi i) \delta(p^2-m^2)$).
Sum over all cuts.
This gives $\operatorname{Im}$ (I am dropping factors ...
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What are the restrictions on the conformal dimension of a primary operator?
Consider a chiral quasi-primary field in 2D with conformal dimension $h$. Under conformal transformation it transforms as:
$$\phi^{'}(z)=(\frac{df}{dz})^{h}\phi(f(z)).$$
For the two point function of ...
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Irreducible representations doubts
I am reviewing irreducible representations and need some clarifications.
Most of what I am writing is based on Schwartz's book, Chapter 8. I will try to use as little group theory as possible, since I ...
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Axioms of quantum mechanics - intuitive explanation
Let me start off by saying that I am a math student (with very little knowledge in physics let alone in quantum mechanics). At the moment I am taking a (first) physics course offered by my university ...
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Can anyone please explain how and why Clebsch–Gordan coefficients form an unitary matrix?
The way Clebsch-Gordan coefficients are defined, I am unable to form an unitary matrix whose elements are Clebsch-Gordan coefficients. How do I construct a matrix that represents transformation. And ...
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What does unitarity violation in QFT translates to? [duplicate]
In QM the probability is violated when:
$\int \rho(\vec r,t)\vec dr\neq 1$ for a quantum mechanical system in an arbitrary state $\psi(\vec r,t)$.
In this case we know that $\psi(\vec r,t)$ is the ...
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The consistent histories approach to QM: choosing the right family of histories
In the consistent histories formalism of quantum mechanics, one consider as basic objects projective decompositions of the identity (PDI). In the following we write a projector as $\left[\phi\right]:=\...
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Cutkosky cutting rules in many-body theory
I have read about the Cutkosky cutting rules and optical theorem when I was studying for theoretical particle physics. I.e. Imaginary part of Greens function is directly correlated to the sum of decay ...
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Why do we need (or prefer?) $SU(2)$ rather than $U(2)$ to perform rotation of 3d Pauli vectors?
In order to rotate the usual 3d vectors (Written as Column vectors), We start with the idea that Rotation perserves lengths, which leads us to the group of $O(3)$. But Reflections also perserve length,...
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Differential Inelastic Cross section and Optical Theorem
Context
For a single channel scattering problem, it is well known that we have to solve the wave function from the Schrodinger equation:
\begin{equation}
\left( -\frac{1}{2m} \nabla^2 + V(r) \right) \...
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Transformation law of quantum fields
I have a short question about the transformation laws of quantum fields. Although I have come across similar questions, I haven't been able to understand this explicitly.
In a QFT, we postulate that ...
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Lagrangian with a negative kinetic term
The Lagrangian density of a canonical scalar field is
$$
L=-\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-V(\phi)
$$
if we use a $(−,+,+,+)$ sign convention.
If the sign of the kinetic term is ...
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