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The transition amplitude for a particle to go from |x> to |x'> isThinking again about the math of virtual particles. They are supposed to come in pairs, but together they don't result in anything permanent. What kind of math would do that? What kind of math leads to complete annihilation or cancellation for two particles that both start at the same point at the same time and both end at the same different point at the same time? ...

[tex] < x'|U(t)|x > \,\,\, = \,\,\,{\left( {\frac{m}{{2\pi \hbar it}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{im{{(x' - x)}^2}/2\hbar t}}.[/tex]

I take this as true even for a virtual particle. The antiparticle is said to travel backwards in time between the same two points. So its transition amplitude would be

[tex] < x|U(t)|x' > \,\,\, = \,\,\,{\left( {\frac{{ - m}}{{2\pi \hbar it}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{ - im{{(x - x')}^2}/2\hbar t}}[/tex]

by simply replacing

*t*with

*-t*. (Or take the complex conjugate). Yes, you can argue that these transitions are not measurable at these specific points since the |x> basis is a continuous spectrum. Granted! But bear with me because I'm trying to prove just that. I'm just considering the transition from some generic point to another generic point for a virtual particle pair that is said to go from some point to another.

The minus sign comes out of the square-root as the complex number

*i*. So, we get

[tex] < x|U(t)|x' > \,\,\, = \,\,\,i{\left( {\frac{m}{{2\pi \hbar it}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}

\kern-0.1em/\kern-0.15em

\lower0.25ex\hbox{$\scriptstyle 2$}}}}{e^{ - im{{(x - x')}^2}/2\hbar t}}[/tex]

This is not a transition from one point by a particle and then back again by the antiparticle. The two transitions happen at the same time. And a real particle might interact with one or the other, and we can't say which. So we can consider these two virtual particles to be in superposition with each other. And then the expectation value for measuring these particles would be

[tex]| < x'|U(t)|x > {|^2}\,\, + \,\,\,| < x|U(t)|x' > {|^2}[/tex]

As seen from the above, the only difference between these terms is the complex number

*i*in the antiparticle. After squaring it the only difference would be a minus sign, and the sum would be zero. This is what we are told, that they exist as wave functions but have zero expectation value of ever being measured. So you could fill space with as many virtual particle pairs as you like, even infinitely many, and it would not be noticeable by any observer.

Did I get my math right?

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