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Field Quantization Without Divergences

Field Quantization Without Divergences. John R. Klauder University of Florida Gainesville, FL. Dirac on Divergences.

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Field Quantization Without Divergences

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  1. Field Quantization Without Divergences John R. Klauder University of Florida Gainesville, FL

  2. Dirac on Divergences Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it! .

  3. Frequently Asked Questions • No divergences. Is that possible? YES • What is the ‘‘cost’’? AN UNUSUAL LOCAL COUNTER TERM • Basic strategy? SOLVE NONRENORMALIZABLE CASES • Which fields? SCALARS (HIGGS), [GRAVITY, FERMIONS]

  4. Outline • Background (Scalar Fields); Basic Proposal • Free/Pseudofree Models • Why Divergences Arise • How Divergences Appear • Relevance for Scalar Fields • The Cure: ‘‘Measure Mashing’’ • Lattice Hamiltonian • Lattice Action • Monte Carlo Evidence • Other Fields • Origin of Measure Mashing

  5. Background (Scalar Fields)

  6. Basic Proposal

  7. Free/Pseudofree Models F CON. g

  8. Free/Pseudofree Models PF DISCON. F CON. AN “AMPUTATED” ACTION FUNCTIONAL g

  9. Scalar Fields B.I. PF NONREN. F REN. g o AN “AMPUTATED” ACTION FUNCTIONAL

  10. Why Divergences Arise B.I.

  11. Many Variables

  12. Support when N = Infinity

  13. How Divergences Appear

  14. Relevance for Scalar Fields

  15. Ground State Distribution WILL THE GUILTY VARIABLES THAT LEAD TO DIVERGENCES PLEASE RAISE THEIR HANDS

  16. Ground State Distribution

  17. Moments in the New Coordinates

  18. Moments in the New Coordinates

  19. Moments in the New Coordinates

  20. The Cure: ‘‘Measure Mashing’’ B.I.

  21. Counter Term s=2

  22. Lattice Hamiltonian B.I.

  23. Spacetime and Space Averages

  24. Lattice Action

  25. Monte Carlo Evidence

  26. g_R vs. g_0 for n=3 and n=4

  27. Phi^4_4 With Counter Term

  28. Other Fields

  29. Origin of Measure Mashing

  30. Ultralocal Scalar Fields

  31. Summary • Lattice ground state wave function ‘‘=’’ Lattice Hamiltonian‘‘=’’Lattice action • Origin of divergences traced to power of the hyper-spherical radius • Measure mashing changes mutuallysingular measures into equivalent measures • Finite spatial moments implies finite spacetime moments • Monte Carlo supports non-triviality

  32. Feynman on Divergences . The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate.

  33. Thank You

  34. References • ``Scalar Field Quantization Without Divergences In All Spacetime Dimensions'' J. Phys. A: Math. Theor.44, 273001 (2011); arXiv:1101.1706 • ``Divergences in Scalar Quantum Field Theory: The Cause and the Cure'', Mod. Phys. Lett. A27, 1250117 (9pp) (2012); arXiv:1112.0803 • Ultralocal model scalar quantum fields: ‘‘BeyondConventional Quantization’’ (Cambridge, 2000 & 2005) • ‘‘Recent Results Regarding Affine Quantum Gravity’’, J. Math. Phys. 53, 082501 (19pp) (2012); arXiv:1203.0691

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