Applying General Lorentz Boost to Multipartite Quantum State

[Emil_M]

TL;DR Summary

How to find unitary transformation corresponding to a general Lorentz transformation, that will perform a change of reference frame on a multipartite quantum state

I would like to apply a General Lorentz Boost to some Multi-partite Quantum State.

I have read several papers (like this) on the theory of boosting quantum states, but I have a hard time applying this theory to concrete examples.

Let us take a $|\Phi^+\rangle$ Bell State as an example, and apply a general Lorentz Boost $$
\Lambda=\left[\begin{array}{cccc}{\gamma} & {-\gamma \beta_{x}} & {-\gamma \beta_{y}} & {-\gamma \beta_{z}} \\ {-\gamma \beta_{x}} & {1+(\gamma-1) \frac{\beta_{x}^{2}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{x} \beta_{y}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{x} \beta_{z}}{\beta^{2}}} \\ {-\gamma \beta_{y}} & {(\gamma-1) \frac{\beta_{y} \beta_{x}}{\beta^{2}}} & {1+(\gamma-1) \frac{\beta_{y}^{2}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{y} \beta_{z}}{\beta^{2}}} \\ {-\gamma \beta_{z}} & {(\gamma-1) \frac{\beta_{z} \beta_{x}}{\beta^{2}}} & {(\gamma-1) \frac{\beta_{z} \beta_{y}}{\beta^{2}}} & {1+(\gamma-1) \frac{\beta_{z}^{2}}{\beta^{2}}}\end{array}\right]
$$ to this state.

Now, as I understand, we represent this Lorentz Boost as some unitary $U(\Lambda)$ in our Hilbert Space, in order to be able to boost our quantum state:$$|\Phi^{+'}\rangle=U(\Lambda)|\Phi^+\rangle$$

Unfortunately, I have found no paper that detailes just how exactly this unitary is found, they all simply state that it must always exist.

So, how would I find $U(\Lambda)$ that boosts some quantum state – like $|\Phi^+\rangle$ – from some inertial frame of reference $S$ to another $S'$?

Thanks!

 

[HomogenousCow]

It depends on the Hilbert space you’re working with. If it’s a Fock space then in the momentum basis the representation of the lorentz boosts simply boost the three momentums the usual way. If your particles have spin it’s more complicated and you need to boost the polarization states as well. There are partial discussions in Peskin and Schroeder for the spin-0 and 1/2 cases.

 

[meopemuk]

@HomogeneousCow: Your described action of boosts on the momenta and spins of particles is valid only in systems of non-interacting particles. When particles interact with each other, then the boost generator becomes interaction-dependent (in a sense, similar to the interaction-dependent Hamiltonian).
This is true for interacting dynamics in the instant form, as explained in

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949), 392.

Eugen.