Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis
Abstract
The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For KleinGordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for KleinGordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positiveenergy solutions. We show that this problem is solved very naturally in complex spacetime, where $f(xiy)^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinorvalued relativistic coherent states. For free quantized KleinGordon and Dirac fields, the above formalism extends to nparticle/antiparticle coherent states whose scalar products are Wightman functions. The 2point function plays the role of a reproducing kernel for the oneparticle and antiparticle subspaces.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 DOI:
 10.48550/arXiv.0910.0352
 arXiv:
 arXiv:0910.0352
 Bibcode:
 2009arXiv0910.0352K
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Quantum Physics;
 32XX;
 35XX;
 41XX;
 44XX;
 46XX;
 53XX;
 81XX;
 83XX
 EPrint:
 252 pages, no figures. Originally published as a book by NorthHolland, 1990. Reviewed by Robert Hermann in Bulletin of the AMS Vol. 28 #1, January 1993, pp. 130132