Operator Ordering Conventions and Symmetry

Question

Quantization procedures may need operator ordering conventions to avoid ambiguity. In classical theories, classical observables are often described by smooth functions, so the order of observable products is irrelevant. In the quantum setting, quantum observables are operators, so interpreting products of observables may need some ordering convention.

I would like to ask:

1. Are there any operator orderings which are more natural than others?
2. What justifications are used to suggest certain ordering conventions are more natural than others?
3. How is the 'naturality' of certain ordering conventions related to symmetry?

For instance time ordering, path ordering and radial ordering all seem to be 'natural' operator ordering procedures in certain situations.

I've heard people say the naturality of these choices depends on the symmetries within the system. For instance, a relativistic system may be naturally viewed as a foliation of equal-time slices. Similarly, systems with dilation symmetry (like CFTs) may naturally be foliated via radial slices.

From another perspective, I know divergences sometimes arise because a classical symmetry fails to quantize. However I've also heard that certain operator orderings can lead to anomaly cancellation (e.g. normal ordering in canonical quantization leading to vacuum energy anomalies vanishing). Are these operator orderings somehow related to foliations constructed using the same symmetry which originally caused the anomaly?

Alternatively, perhaps there are also physical reasons certain operator ordering conventions are 'natural'. As an example, time-ordering seems to have a causal justification. Similarly, normal ordering can avoid divergences due to vacuum energy.

Am I misunderstanding what I've heard? Can anyone elaborate on these ideas?

Answer 1

1. There is not just 1 natural operator ordering. Instead there a variety of operator orderings in different situations that serve different purposes and with different pros & cons.

   Examples:

   • The path integral formulation typically relies on time-slicing.

   • Time-ordered correlator functions are related to Euclidean formulation via Wick rotation.

   • The choice of operator ordering and choice of vacuum states are interrelated, e.g. a Fock vacuum are interrelated with normal ordering, cf. e.g. my Phys.SE answer here.

   • Weyl/symmetric ordering ensures that the operator is Hermitian.

2. Symmetry is an important principle in limiting the ambiguity of quantization. The ambiguity of quantization is not removed by (and should in principle be independent of) operator ordering, especially if we can go to other operator orderings using Wick's theorem. (Of course, the final formula may look neater in certain choices of operator ordering.)