The Weyl transformation^[1] of a classical function h(p, q) is defined as

which is useful in quantum statistics and path integral.^{[2, 4]} When h(p, q) is quantized as the operator function H(P, Q) through the Weyl correspondence

where Δ (p, q) is the Wigner operator in the coordinate eigen-vector | q⟩ representation^[3]

then the matrix element of H is just the Weyl transformation

which has been used in the mid-point path integral theory.^[4] An interesting question arises: is there a generalized Weyl transformation which can include Weyl quantization and P– Q quantization as well as Q– P ordered quantization as a whole? The answer is affirmative. In this work, we shall extend the above formalism to s-parameterized Weyl transformation, i.e., we extend formula (1) to

where s is an arbitrary real number, and h_s(p, q) is another classical corresponding function of the same H(P, Q) and to be determined.

In Section 2, we derive the s-parameterized Wigner operator. Then in Section 3, we present an s-parameterized quantization scheme which includes P– Q quantization, Q– P quantization, and Weyl ordering as its three special cases. At this point we emphasize that our s-parameterized quantization scheme is quite different from the scheme that includes Weyl quantization and normally ordered quantization as well as anti-normally ordered quantization as a whole. In Section 4, we derive the relation between the s-parameterized classical correspondence and the P representation of operators.

Using the inner product ⟨ q′ | q⟩ =δ (q− q′ ), we can reform Eq.  (2) as

from which we can extract

Then

and

embody a generalized Weyl quantization scheme, so h_s(p, q) is considered as a generalized Weyl classical correspondence of H(P, Q). Using the relation P| q⟩ = id| q⟩ /d_q, we can rewrite Eq.  (7) as

In particular, when s = 0, it reduces to Eq.  (2) as expected. When s = 1, by applying the Baker– Hausdorff formula to Eq.  (10), we have

However, when s = − 1, Ω _s(p, q) becomes the following operator function:

Using Eq.  (7), we can calculate

With the aid of Eqs.  (9) and (13), we can further derive

As an example, when H(P, Q) = Q^mP^r,

Using the completeness relation of the momentum eigenvector

we have

Substituting it into Eq.  (15) yields the s-parameterized classical correspondence of Q^mP^r

In particular, when s = − 1, on the right-hand side only the term of j = 0 survives, so equation  (18) reduces to

which coincides with Eqs.  (9) and (12)

On the other hand, when s = 1, equation  (18) becomes

which along with Eqs.  (9) and (11) indicates

which is the operator identity of rearranging Q^mP^r as P– Q ordering. For instance, for m = r = 1, equation  (22) becomes QP= PQ + i (we have set ħ = 1).

Now we examine the s-parameterized classical correspondence of

Substituting Eq.  (17) into Eq.  (23) leads to

For the special case of s = 0, only the term of l = 0 survives, so

which is just the Weyl correspondence. Thus we obtain the generalized s-parameterized quantization scheme rule

For the special case of s = 1,

while for s = − 1,

these are two new operator identities.

Let and . Using the technique of integration within an ordered product of operators^{[5, 8]} and the following integration formula

with the convergent condition

we can reform Eq.  (9) as

Then using the P-represenation 𝔓 (β ) of H(P, Q) ^{[9, 10]}

we have

which is the relationship between H(P, Q) ′ s P-represenation and its s-parameterized classical correspondence.

In summary, based on the s-parameterized Wigner transformation, we have derived the s-parameterized quantization scheme, which includes P– Q quantization, Q– P quantization, and Weyl ordering as its three special cases and is useful to derive some new operator identities. For the formula converting operators into s-parameterized ordering, we refer to Ref. [11].