Good Curses

Cosma Shalizi's review of Physics from Fisher Information by Roy Frieden.

Hans Moravec's review of Shadows of the Mind by Roger Penrose.

Concerning Barbour's paradox [the elimination of time in GR] see this dialogue.

This page, Good Curses, is dedicated to blistering reviews and blustering views. One of these terms applies to Lubos Motl's review of our book, PCT, Spin and Statistics and All That. To judge which, go to amazon.com and search for "Streater, Wightman" in the book products. Click on "New and Used"; and "product details" and then you will be shown the reviews. Amazon invites you to vote whether you find this review helpful or not. To assist you in this decision, consider Motl's claim that we say that it is possible to define a parity operator in any theory. This remark does not pass the giggle test^*. He mentions solitons as one of the topics omitted from the book. He obviously gave up reading it before he got to the appendix. Motl also mentions that constructive qft would be a better thing to study than our book. This reveals that he did not read the 12 pages in the Appendix devoted to this subject. Our book is a "girding of the loins", without which constructive qft, which started with Wightman's 1964 article, would not have flourished. Even string theory owes its development to some extent by the analysis of conformal qft models; they were proved to satisfy Wightman's axioms, and so their meaning was unambiguous.

Motl's main claims are that Wightman theory has given no physical predictions, and that nevertheless "most of its conclusions are believed to be incorrect". PCT incorrect? Spin and statistics incorrect? Have I missed some important recent experiments? He concludes that "the number of incorrect and misleading sections is too large". Let us know what these are, Lubos, and we shall do our best to correct them for the next edition (and all further editions).

The prehistory of Motl's review can be traced to a chat site, physicstalk. Someone hiding under the nom-de-guerre LR misread a simple theorem in our book, and asked the bloggers to explain our "claim" that all theories must be invariant under parity. Motl and others replied that this was not true in models; Motl concluded that the theorem must be false, not from a misreading of the book, but from a non-reading. See the sites
here
and
here

The nub of the confusion can be seen in one of LR's blogs. He says that we claim to show that if one can define a transformation law of the field under parity, then parity is a symmetry. Rather, he has in mind a substitution law, in which symbols for the field components are rearranged to linear sums of such symbols at a transformed space-time point. The distinction between this formalism, used by some physicists and in particular Schwinger, and the requirement of the existence of a unitary or anti-unitary operator on the states implementing the substitution law, is lost in the word "transformation". In the Wigner formulation, we need to implement the transformation law. More, the same unitary operator (for parity) must implement the transformation of the fields phi(t,x) for all time.

Earlier in the book, on page 16, we do warn the reader of this distinction: "For clarity, we want to recall the relation between our description of symmetries and that which was standard in the quantum field theory of 30 years ago. Then a symmetry was given as a substitution law, A maps to A' for the operators A in terms of which the theory was formulated ... Symmetry was guaranteed by requiring that the equations of motion be form-invariant under the substitution...It was tacitly assumed that to each substitution the exists a unitary transformation Phi mapsto Phi' ...[of the states]". Thus, it is no use just doing a substitution: we must say what happens when we do it. In the path-integral formalism, the Lagrangian must be invariant; in the Hamiltonian formalism, the equations of motion must be invariant. And in Wigner's formalism, the unitary operator must exist, and do the job for fields at any time.

The substitution rules given on page 19, eq. (1-45) and page 20, (1-47) and (1-50), do not specify a space-time point. For parity, it is (t,x) on the left and (t,-x) on the right. The substitution rule might not make sense if there is no unitary operator doing the job. In the Hamiltonian formalism, the fields at a later time are functions of the fields (including the conjugate fields) on any previous time-slice. Thus, making the substitution at time zero fixes how the fields transform at a later time. If parity is not conserved, so that any U(I_s) depends on time, this will be different from that implied by the substitution rule at a later time. Then the transformation law is inconsistent and does not exist as a transformation of the operators built out of field operators. In the path-integral formalism, the fields are taken as classical variables (Grassmann variables in the case of Fermions), not related by any equation of motion. In this case, the Lagrangian must be invariant, to get a symmetry. When these three formalisms all make rigorous sense, as in constructive qft, then the first two notions are equivalent, but Wigner's requirement is stronger. In other words, there exist models (of spontaneous breakdown of symmetry) in which a substitution rule leaves the Lagrangian and the equations of motion invariant, but there is no unitary operator doing the job. Thus the advice to LR given by Motl and the other bloggers was not accurate. See my paper Spontaneous Breakdown of Symmetry in Axiomatic Theory. This can sometimes be expressed by saying that the vacuum is not invariant under the dual of the transformation, or that the measure in the functional integral is not invariant. I am endebted to Gerry Guralnik for putting me onto this question.