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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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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Functional equation (6.156) in the book "QFT" by Lewis Ryder
I have 2 questions regarding Eq. $6.155$ to Eq. $6.156$ in Ryder's QFT book. In Eq. $6.155$, why do we multiply the functional $I[J]$ from the left of $\phi_{in}(x)$ and then from the right of $\phi_{...
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Generating entangled states with a unitary operator
I'm trying to find a unitary operator U to take an initial, unentangled tensor product state and generate an entangled state where each basis state of the first subsystem is uniquely correlated with ...
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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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Charge number and current operators in Heisenberg picture
By proper definition of the unitary operator $\hat{U}$ we can have the time-evolution of an operator $\hat{Q}^H$ (superscript $H$ for Heisenberg picture; $\hat{Q}^H=\hat{U}^\dagger \hat{Q}\hat{U}$) as:...
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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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The meaning of $i$ in the Yang-Mills
In Yang mills theory (including maxwell's theory), the generators of $SU(n)$ are complex matrices acting on the spinor $\psi$. Now the phase of $\psi$ is also given by $e^{i\theta}$. Why do we use ...
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Proving that a Hamiltonian transformation is canonical - Condensed Matter, Altland and Simons
In the book Condensed Matter Field Theory of Altland and Simons, page 63 eq 2.30 in the second edition, they are doing the following transformation for the Hamiltonian: $$\hat{H} \rightarrow \hat{H'} ...
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Connection between Real time evolution and Imaginary time evolution
Let $H = P$ be the Hamiltonian described by a Pauli operator $P$. The real time evolution according to H is $$e^{-iPt}.$$ While the imaginary time evolution is $$e^{-Pt}.$$ Consider the case $P=Y$. ...
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Time evolution and anti-unitary operators
Let $\hat{H}$ be a system of lattice fermions with two internal states per site, so they can be described using operators $\hat{c}_{m,\alpha}$, with $m$ denoting the lattice site and $\alpha$ the ...
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Transformations on correlation functions
Let $\hat{U}$ be a unitary transformation that acts on fermionic fields (defined on a lattice) as $$\hat\psi_j \rightarrow \hat{U}\hat\psi_j\hat{U}^{-1},$$ $$\hat\psi_j^\dagger \rightarrow \hat{U}\hat\...
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Is there a no-cloning theorem for open quantum dynamics?
There are numerous papers that prove a no-cloning theorem (or more generally a no-broadcasting theorem) at various levels of generality. However, it is unclear to me if Cloning is proven to be ...
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Linkage of a classical canonical transformation to a quantum unitary transformation
Suppose I have an classical Hamiltonian $H(q, p)$ and the quantum equivalent $\hat{H}(\hat{q}, \hat{p})$. We can apply linear canonical transformation defined by a type-II generating function $S_2(q,p^...
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How to maximally entangle two qubits? [duplicate]
I'm looking for a transformation that entangles two qubits $\mid{\psi}_1\rangle = \alpha_1 {\mid}0\rangle + \beta_1 {\mid}1\rangle$ and $\mid{\psi}_2\rangle = \alpha_2 {\mid}0\rangle + \beta_2 {\mid}...
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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 &...
