Renormalized oscillator equations

Abstract

Let {H=L2(R,dx), q=x,p=−id/dx} be the standard Schrödinger representation of the canonical commutation relations for one degree of freedom. For a, b, c, real scalars, put Habc = ½p2 + q4 + aq3 + bq2 + cq and q(t) = exp(itHabc)q exp(− itHabc). Then q(t) satisfies the operator equation q̈(t) = − 4q3(t) − 3aq2(t) − 2bq(t) − c. For n = 0, 1, 2, 3 we define renormalized (or Wick-ordered) nth powers of q, denoted q(n); these are polynomials of degree n in q and are characterized by the conditions: {q(0)=1 [q(n+1),p]=i(n+1)q(n)for n>0,∫G*q(n)Gdx=0,where G=ground state of Habc}. In terms of these powers the equation for q(t) can be written q̈(1) = − 4q(3) − Aq(2) − Bq(1) − C, for some real A, B, C which are functions of a, b, c. This has the form of a much studied prototypical nonlinear quantum field equation in the somewhat (physically) trivial case of one space-time dimension. Basically, we prove two results concerning this equation which we feel are of interest, because they may provide some basis for conjecture about the behavior of nonlinear field equations in a higher number of dimensions. First we determine a nontrivial condition which the renormalization constants A, B, C must satisfy and which implies that the set of points (A, B, C) assumed as a, b, c vary is of measure zero in R3. We study in somewhat more detail the situation when a = c = 0. The associated renormalized equation then has the form q̈(1) + Bq(1) = − 4q(3), where B = B(b). We determine the qualitative behavior of the function B(b) as b → ± ∞ and show for example that B(b) is not one-to-one. As a corollary, for many values of B, there exist at least two equations of the form q̈(1) + Bq(1) = − 4q(3) with the same B, but which are not unitarily equivalent in a sense, to be precised. Such nonunicity cannot occur for linear equations, as has been known for some time.