Mandl and Shaw 2.5

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[SOLVED] Mandl and Shaw 2.5

The question is to show that the unitary transformation corresponding to spacetime translation [itex]\delta_\alpha[/itex] of a scalar field is [itex]U = e^{-(\mathrm{i}/\hbar) \delta_\alpha P^\alpha }[/itex] where [itex]P^\alpha[/itex] is the energy-momentum 4-vector of the field.

[itex]\varphi (x) \mapsto \varphi'(x') = \varphi(x_\alpha - \delta_\alpha) = U\varphi(x)U^\dag[/itex].

Essentially this boils down to showing that

[itex]\varphi(x_\alpha-\delta_\alpha) = U \varphi(x_\alpha)U^\dag[/itex].

I'm sure I need to use the identity

[itex][P^\alpha, U] = -\mathrm{i}\hbar\frac{\partial U}{\partial x_\alpha}[/itex],

but I'm not sure how to contort it into a form that will give me what I want.
 

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  • #2
George Jones
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I think the idea is to use

[tex]U \psi \left(x\right) = \left[ U , \psi \left(x\right) \right] + \psi \left(x\right) U[/tex]

to write

[tex]U \psi \left(x\right) U^{\dagger} = \left[ U , \psi \left(x\right) \right] U^\dagger + \psi \left(x\right).[/tex]

Then use

[tex] \left[P^a , \psi \left(x\right) \right] =-i \hbar \frac{\partial \psi}{\partial x^a} \left(x\right)[/tex]

and the power series expansion of [itex]U[/itex] to work out

[tex]\left[ U , \psi \left(x\right) \right].[/tex]
 
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  • #3
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Interesting idea but I'm not sure if it can be made to work,

[itex][P^\alpha,\varphi] = -\mathrm{i}\hbar\partial^\alpha \varphi \implies [P^\alpha P^\beta,\varphi] = -\mathrm{i}\hbar(P^\alpha \partial^\beta\varphi + \partial^\alpha \varphi P^\beta)[/itex].

Higher terms in the expansion of [itex]e^{-\mathrm{i}\hbar \delta_\alpha P^\alpha}[/itex] contain bigger and bigger versions of this.
 
  • #4
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Scrap that last post, the point is when the displacement is infinitesimal, higher order terms don't contribute. Finite displacements can be obtained by infinitely compounding the infinitesimal operator.
 
  • #5
George Jones
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Scrap that last post, the point is when the displacement is infinitesimal, higher order terms don't contribute. Finite displacements can be obtained by infinitely compounding the infinitesimal operator.
I think that by playing with commutators, the full Taylor series expansion of [itex]\varphi(x_\alpha-\delta_\alpha)[/itex], can be obtained, not just the first two terms of the Taylor series expansion.

In the middle, things probably get somewhat messy, though.
 

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