# Arthur Wightman

Arthur Strong Wightman (March 30, 1922 – January 13, 2013) was an American mathematical physicist, known for the Wightman axioms.

## Quotes

* Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from Lorentz invariance, the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.
** (1956). "Quantum Field Theory in Terms of Vacuum Expectation Values". Physical Review 101 (2): 860–866. DOI:10.1103/PhysRev.101.860.
* … there are other things wrong with these models but the fundamental trouble is the non-uniqueness of the vacuum as was first shown by HEPP, JOST, RUELLE and STEINMANN … Actually, BORCHERS … has shown that the cluster decomposition property is not only necessary but sufficient for the uniqueness of the vacuum, if there is at least one cyclic vacuum. … HEPP, K., JOST, R., RUELLE, D. and STEINMANN, O., Necessary condition on Wightman functions, Helv. Phys. Acta 34 (1961) 542. BORCHERS, H.J., On the structure of the algebra of field observables, Nuovo Cim. 24 (1962) 214
** (1963). "Recent achievements of axiomatic field theory". Theoretical Physics: 11-58. (quote from p. 30)
* Why was the discovery of the Dirac equation ... so important for the physics of the 1920s and 30s? The answer is manifold. The Dirac equation provided a relativistic description of spin ½ particles and in particular of the electron. In doing so, it gave a relativistic description of spin and opened the way for the application of group theory to the description of particles of arbitrary spin. The reinterpretation of the Dirac equation as a field equation that followed from Dirac's theory of holes was decisive in the conceptual transformation of single particle theory to many particle (quantum field) theory. The resulting quantum electrodynamics of spin ½ particles, refined by two generations of theoretical work, is the best theory we have. Although it is an approximation since it does not include the effects of weak and strong interactions it has survived many stringent experimental tests when applied to electrons and muons.
** Salam, Abdus & Wigner, E. P., eds. (2 November 1972). "Chapter 6. The Dirac Equation by A. S. Wightman". Aspects of Quantum Theory (volume of essays in honour of Paul Adrien Maurice Dirac). Cambridge University Press. pp. 95–116. ISBN 9780521086004. (quote from p. 95)
* The family of mathematical problems discussed here has emerged in recent years as a result of efforts to put a small chapter of quantum field theory, the so-called external field problem, on a sound mathematical footing. The external field problems is special because the partial differential equations for the unknown field is linear, but the coefficients are allowed to vary in space and time and that gives rise to some surprises, which seem to be of general interest. There is a vast and in large part turgid mathematical physics literature on the subject. To make the general wisdom which has accumulated there more readily available to a mathematical audience I have, in the following, tried to place the problems in their physical context, and still to bring out the essential mathematical questions many of which remain to be answered.
** "Relativistic wave equations as singular hyperbolic systems by A. S. Wightman". Partial Differential Equations. Proceedings of Symposia in Pure Mathematics. 23. 1973. pp. 441–477. doi:10.1090/pspum/023/0342075. ISBN 9780821814239. (quote on p. 441)
* From the very beginning of quantum mechanics, the notion of the position of a particle has been much discussed. In the nonrelativistic case, the proof of the equivalence of matrix and wave mechanics, the discovery of the uncertainty relations, and the development of the statistical interpretation of the theory led to an understanding which, within the inevitable limitations of the nonrelativistic theory, may be regarded as completely satisfactory.
** "On the Localizability of Quantum Mechanical Systems". Special Relativity and Quantum Theory. 1988. pp. 402–429. doi:10.1007/978-94-009-3051-3_37. ISBN 978-94-010-7872-6.

## Quotes about Arthur Wightman

* ... when I was a graduate student in the 1950s, I had to learn a lot about cosmic ray physics — because that's where all the information was coming from about new particles. And I remember how surprised I was when a professor — at Princeton where I was a student — Arthur Wightman, told me that pretty soon physicists would no longer be worried about cosmic rays. They would be getting the information about particles from new kinds of accelerators — which would accelerate known particles like protons, which are the nuclei of hydrogen atoms, or alpha particles to very high energy where they would collide with each other or with stationary targets. And in that collision new matter would be formed.
** Steven Weinberg, (February 20, 2015)"Steven Weinberg: On the Shoulders of Giants". World Science Festival, YouTube. (original program date: June 4, 2011; quote at 9:14 of 1:01:20)

## External links

Encyclopedic article on Arthur Wightman on Wikipedia