Vector Affine Quantization Can Create Valid Quantum Field Theories

Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically ( ) 2 D PQ QP = + and 0 Q ≠ , to represent appropriate kinetic factors, such as 2 P , which involves only one canonical quantum operator. The need for this requirement stems from path integral quantizations of selected problems that affine quantization can solve but canonical quantization fails to solve. This task is resolved for simple examples, as well as examples that involve scalar, and vector, quantum field theories.


Elements of Basic Affine Quantization
Affine quantization (AQ) has been created from canonical quantization (CQ) in the sense that from P and Q, which obey [ ] , 1l  Recently, the affine version of the kinetic operator 2 P , which is , was analyzed. It was shown that it led to a different correction term which is proportional to 2  . While the classical Hamiltonian for the harmonic oscillator is ( ) The vector version, which can retain rotational symmetry of this quantum Hamiltonian, would be ( ) In addition, this expression can also require that every oscillator has only positive coordinates. For a two-component vector, we can imagine that accepted vectors resemble a clock's minute hand which only points between 12 noon and 3 pm.

A Scalar Model That Is Familiar
Our first model has a classical Hamiltonian given by  is the interaction power, and 1 n s = + is the number of It follows that the classical Hamiltonian in affine variables is given by , and it all leads to require that By just using AQ variables, instead of CQ variables, it is noteworthy that Equation (2) has absolutely eliminated any non-renormalizability! Using AQ and Schrödinger's representation, the quantum Hamiltonian for Journal of High Energy Physics, Gravitation and Cosmology this model is which, following Section 1, leads to a formal version of the affine quantum Hamiltonian, At this point we introduce a paragraph from [4] that shows how to eliminate , where a is a tiny spacial distance between neighboring lattice points. In preparation for an integration, just as every integral involves a continuum limit of an appropriate summation, these expressions are used in Monte Carlo (MC) calculations which involve proper sums for their 'integrals'. All of this is designed to provide a path integral quantization, and, when necessary, their sums need to be regularized. In our case, the regularized version becomes appropriately 'scaled': specifically In this paper, there will be other models that introduce ( ) 2 0 s δ type divergences. The kind of procedures outlined in the foregoing paragraph can tackle any one of them.

An Example in Which AQ Passes to CQ
To illustrate this example with CQ, we first choose The regularized sum in (5) has been scaled in which the factor

Two Valid Affine Quantizations of Vector Models
Our first task will be to turn a scalar field model into a vector field model. The classical Hamiltonian in (1) is our first target. All that is necessary is to let for two of the scalar terms, and

A Vector Model That Is Common
The classical Hamiltonian for this vector model is The term 1 n s = + represents the number of spacetime dimensions as before, while now 2,3, 4, r =  , is the interaction power. Such models can also fail with CQ, and we will focus on AQ. The classical dilation variable now is . The quantum Hamiltonian, expressed in affine variables and in Schrödinger's representation, is given by In this case the kinetic factor, again with Schrödinger's representation, becomes and once again a scaled version to eliminate ( ) 2 0 s δ is readily obtained. By choice, this vector model exhibits full rotational symmetry.

A Vector Model That Is Less Common
The classical Hamiltonian, expressed in canonical variables, is given by ( )