Nonlinear quantum fields in ≥4 dimensions and cohomology of the infinite Heisenberg group (1994)

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the free massless scalar field in <italic>d</italic>-dimensional Minkowski space (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, even) and if <italic>q</italic> is an even integer greater than or equal to 4, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript bold upper M 0 Endscripts colon phi left-parenthesis upper X right-parenthesis Superscript q Baseline colon d upper X"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo largeop="false">∫<!-- ∫ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">M</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>d</mml:mi> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper X right-parenthesis Superscript q"> <mml:semantics> <mml:mrow> <mml:mi>ϕ<!-- ϕ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi {(X)^q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals q equals 4"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d = q = 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to corresponding scattering theory.