- #1

Emil_M

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- 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!

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!