Wednesday, 8 October 2014

What is the analogy of |xrangle in quantum field theory?


Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have U(xb,xa;T)=xb|U(T)|xa=DqeiS

|xa is an eigenstate of position operator ˆx.


In QFT we have U(ϕb,ϕa;T)=ϕb|U(T)|ϕa=DϕeiS

|ϕa is an eigenstate of field operator ˆϕ(x).


By analogy with QM, it is tempting to relate |ϕ|x


However, in Peskin and Schroeder's QFT, p24, by computing it is said




0|ϕ(x)|p=eipx

We can interpret this as the position-space representation of the single-particle wavefunction of the state |p, just as in nonrelativistic quantum mechanics x|peipx is the wavefunction of the state |p.



Based on the quoted statement, seems ˆϕ(x)|0|x


If relations (3) and (4) are both correct, I should have ˆϕ(ˆϕ|0)=ϕ(x)(ˆϕ|0)

seems Eq. (5) is not correct. At least I cannot derive Eq. (5).


How to reconcile analogies (3) and (4)?



Answer





  1. No ˆϕ|0 is not an eigenvector of ˆϕ. You can see this, for example, by writing out ˆϕ in terms of creation and annihilation operators, then compare ˆϕ|0 against ˆϕ2|0, and observe that one is not a scalar multiple of the other. So as you suspected, eq. 5 is not correct





  2. To obtain some analogy of |x, you can just take a fourier transform of a(p) to get a(x), and a(x)|0|x is the best analogy of |x that I can think of




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