Unitarity, Locality, and Scale versus Conformal Invariance in Four Dimensions

K. Osterwalder and R. Schrader, Commun.Math.Phys. 31

1973

Earlier work this paper cites.

S. R. Coleman, Commun.Math.Phys. 31

1973

Earlier work this paper cites.

Strictly speaking, “time-ordered correlation functions” may be regarded as analytic continuation of Euclidean correlation functions. There, values at x i = x j x_{i}=x_{j} are not defined (even as distribution) in the Osterwalder-Schrader axioms, and these values are unimportant in reproducing Wightman distributions Osterwalder and Schrader 1973 ; Osterwalder and Schrader 1975 . However, one should not misunderstand that the contact terms appearing in Ward identities or anomalies are meaningless. There, we take derivatives (or trace in the case of trace anomaly) of correlation functions and get contributions which have support on x i = x j x_{i}=x_{j} . These contact terms contain information of nonlocal behavior x i ≠ x j x_{i}\not=x_{j} of the original correlation functions before taking derivatives or trace, and therefore meaningful. For example, g μ ​ ν ​ ( p μ ​ p ν / p 2 ) = 1 g^{\mu\nu}(p_{\mu}p_{\nu}/p^{2})=1 is local, but ( p μ ​ p ν / p 2 ) (p_{\mu}p_{\nu}/p^{2}) is not. Our interest in this paper is not the original operator T μ ​ ν T_{\mu\nu} , but T T itself, so it is our freedom to drop contact terms. If we restore those contact terms, our result is written as g μ ​ ν ​ T μ ​ ν = ∂ 2 O + ( contact ​ terms ) g^{\mu\nu}T_{\mu\nu}=\partial^{2}O+({\rm contact~terms}) in time-ordered correlation functions, which is the usual statement of Ward identities for the trace of T μ ​ ν T_{\mu\nu} . We stress that there is no notion of contact terms in the Wightman’s case

1975

Earlier work this paper cites.

K. Osterwalder and R. Schrader, Commun.Math.Phys. 42

1975

Earlier work this paper cites.

A. Zamolodchikov, JETP Lett. 43

1986

Earlier work this paper cites.

J. Polchinski, Nucl.Phys. B303

1988

Earlier work this paper cites.

R. Streater and A. Wightman, (1989)

1989

Earlier work this paper cites.

I. Jack and H. Osborn, Nucl.Phys. B343

1990

Earlier work this paper cites.

H. Osborn, Nucl.Phys. B363

1991

Earlier work this paper cites.

S. Weinberg, (1995)

1995

Earlier work this paper cites.

S. Weinberg, (1996)

1996

Cited alongside, same era.

V. Riva and J. L. Cardy, Phys.Lett. B622

2005

Cited alongside, same era.

B. Grinstein, K. A. Intriligator, and I. Z. Rothstein, Phys.Lett. B662

Original

2008

Cited alongside, same era.

D. Dorigoni and V. S. Rychkov, (2009), arXiv:0910.1087 [hep-th]

Original

2009

Cited alongside, same era.

S. El-Showk, Y. Nakayama, and S. Rychkov, Nucl.Phys. B848

Original

2011

Cited alongside, same era.

I. Antoniadis and M. Buican, Phys.Rev. D83

Original

K. Yonekura, JHEP 1304

Original

2013

Later among the works it cites.

A. Dymarsky, Z. Komargodski, A. Schwimmer, and S. Theisen, (2013), arXiv:1309.2921 [hep-th]

Original

2013

Later among the works it cites.

K. Farnsworth, M. A. Luty, and V. Prelipina, (2013), arXiv:1309.4095 [hep-th]

Original

2013

Later among the works it cites.

O. Aharony, N. Seiberg, and Y. Tachikawa, JHEP 1308

Original

2013

Later among the works it cites.

F. Baume, B. Keren-Zur, R. Rattazzi, and L. Vitale, (2014), arXiv:1401.5983 [hep-th]

Original

2014