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Tagged with unitarityhilbert-space
129 questions
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Topology and continuous families of unitary theories
Let $\mathcal{H}$ be an infinite-dimensional (separable) Hilbert space. In both physics and mathematics, we consider the unitary group $U(\mathcal{H})$, which consists of all unitary operators on $\...
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Is there a structure preserving transformation in quantum mechanics, similar to Lorentz and canonical transformations?
I have noticed how similar the structure preserving property of Lorentz transformations and canonical transformations look, for them we have: $$\Lambda^Tg\Lambda=g, \quad D^TJD=J,$$ with $\Lambda$ ...
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What is the effective dimension of subspace when global unitary is restricted?
Consider any three-qubit state $\vert \psi \rangle \in \mathcal{H}^{\otimes 3}$, and let $U$ be a global unitary that can act on the whole state $\vert \psi \rangle$. Then, it is obvious that the &...
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Unitary Operator action on a state
I have a state $| {\psi} \rangle$ which is acted upon by an unitary operator $e^{ia\hat{x}}$ i.e $$e^{ia\hat{x}}| {\psi} \rangle$$ How does this action take place? Is it similar to the position ...
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Motivation behind reflection positivity
I have taken a look at this Phys.SE question Reflection positivity for general fields, Wikipedia, and this paper Reflection Positivity Then and Now by Jaffe, which go over reflection positivity. ...
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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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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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131views
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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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)$$ where $C$ is the coin operator (acts on spin degree of freedom of ...
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Wigner's theorem for 2-state system
I am trying to see how Wigner's theorem can be proven in a 2-state system. Let, $$ |\psi\rangle= a |0\rangle+b|1\rangle, \quad|\phi\rangle= c |0\rangle+d|1\rangle $$ where all coefficients are complex ...
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Writing a given unitary in the same basis as the Hamiltonian (Operator Representation and Confusion)
I have a simple question concerning how to write the representation of operators, such as unitaries, using a specific order for the basis elements. Let me give you an example. Consider a tripartite ...
- 110
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Can you apply non-unitary operators to a qubit?
I am wondering if it is possible to apply continuous, invertible transformations to a qubit which are not linear, i.e. not elements of $U(N)$ where $N=2^n$ where we have $n$ qubits. Consider $n=1$. ...
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255views
How is the Wigner little group representation of Poincaré group Unitary?
From Weinberg's QFT Vol.1, eq(2.5.11): $$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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