Saturday, July 21, 2018

quantum field theory - single-particle wavepackets in QFT and position measurement


Consider a scalar field ϕ described by the Klein-Gordon Lagrangian density L=12μϕμϕ12m2ϕϕ.


As written in every graduate QM textbook, the corresponding conserved 4-current jμ=ϕiμϕ gives non-positive-definite ρ=j0. If we are to interpret ϕ as a wave function of a relativistic particle, this is a big problem because we would want to interpret ρ as a probability density to find the particle.


The standard argument to save KG equation is that KG equation describes both particle and its antiparticle: jμ is actually the charge current rather than the particle current, and negative value of ρ just expresses the presence of antiparticle.


However, it seems that this negative probability density problem appears in QFT as well. After quantization, we get a (free) quantum field theory describing charged spin 0 particles. We normalize one particle states |k=ak|0 relativistically:


k|p=(2π)32Ekδ3(pk),Ek=m2+k2


Antiparticle states |ˉk=bk|0 are similarly normalized.


Consider a localized wave packet of one particle |ψ=d3k(2π)32Ekf(k)|k, which is assumed to be normalized. The associated wave function is given by


ψ(x)=0|ϕ(x)|ψ=d3k(2π)32Ekf(k)eikx


1=ψ|ψ=d3k(2π)32Ek|f(k)|2=d3xψ(x)i0ψ(x)

.



I want to get the probability distribution over space. The two possible choices are:


1) ρ(x)=|ψ(x)|2 : this does not have desired Lorentz-covariant properties and is not compatible with the normalization condition above either.


2) ρ(x)=ψ(x)i0ψ(x) : In non-relativistic limit, This reduces to 1) apart from the normalization factor. However, in general, this might be negative at some point x, even if we have only a particle from the outset, excluding antiparticles.


How should I interpret this result? Is it related to the fact that we cannot localize a particle with the length scale smaller than Compton wavelength ~ 1/m ? (Even so, I believe that, to reduce QFT into QM in some suitable limit, there should be something that reduces to the probability distribution over space when we average it over the length 1/m ... )




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