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## The Zassenhaus formula and the Pauli matrices

Maple

The Zassenhaus formula and the algebra of the Pauli matrices

Edgardo S. Cheb-Terrab1 and Bryan C. Sanctuary2

(1) Maplesoft

(2) Department of Chemistry, McGill University, Montreal, Quebec, Canada

 The implementation of the Pauli matrices and their algebra were reviewed during 2018, including the algebraic manipulation of nested commutators, resulting in faster computations using simpler and more flexible input. As it frequently happens, improvements of this type suddenly transform research problems presented in the literature as untractable in practice, into tractable.
 As an illustration, we tackle below the derivation of the coefficients entering the Zassenhaus formula shown in section 4 of [1] for the Pauli matrices up to order 10 (results in the literature go up to order 5). The computation presented can be reused to compute these coefficients up to any desired higher-order (hardware limitations may apply). A number of examples which exploit this formula and its dual, the Baker-Campbell-Hausdorff formula, occur in connection with the Weyl prescription for converting a classical function to a quantum operator (see sec. 5 of [1]), as well as when solving the eigenvalue problem for classes of mathematical-physics partial differential equations [2].   To reproduce the results below - a worksheet with this contents is linked at the end - you need to have your Maple 2018.2.1 updated with the Maplesoft Physics Updates version 280 or higher.

References

 [1] R.M. Wilcox, "Exponential Operators and Parameter Differentiation in Quantum Physics", Journal of Mathematical Physics, V.8, 4, (1967.
 [2] S. Steinberg, "Applications of the lie algebraic formulas of Baker, Campbell, Hausdorff, and Zassenhaus to the calculation of explicit solutions of partial differential equations", Journal of Differential Equations, V.26, 3, 1977.
 [3] K. Huang, "Statistical Mechanics", John Wiley & Sons, Inc. 1963, p217, Eq.(10.60).

Formulation of the problem

The Zassenhaus formula expresses  as an infinite product of exponential operators involving nested commutators of increasing complexity

=

Given ,  and their commutator , if  and  commute with ,  for  and the Zassenhaus formula reduces to the product of the first three exponentials above. The interest here is in the general case, when  and , and the goal is to compute the Zassenhaus coefficients in terms of ,  for arbitrary finite n. Following [1], in that general case, differentiating the Zassenhaus formula with respect to  and multiplying from the right by  one obtains

This is an intricate formula, which however (see eq.(4.20) of [1]) can be represented in abstract form as

from where an equation to be solved for each  is obtained by equating to 0 the coefficient of . In this formula, the repeated commutator bracket is defined inductively in terms of the standard commutator by

and higher-order repeated-commutator brackets are similarly defined. For example, taking the coefficient of  and  and respectively solving each of them for  and  one obtains

This method is used in [3] to treat quantum deviations from the classical limit of the partition function for both a Bose-Einstein and Fermi-Dirac gas. The complexity of the computation of  grows rapidly and in the literature only the coefficients up to  have been published. Taking advantage of developments in the Physics package during 2018, below we show the computation up to  and provide a compact approach to compute them up to arbitrary finite order.

Computing up to

Set the signature of spacetime such that its space part is equal to +++ and use lowercaselatin letters to represent space indices. Set also ,  and  to represent quantum operators

 >
 >
 (1)

To illustrate the computation up to , a convenient example, where the commutator algebra is closed, consists of taking  and  as Pauli Matrices which, multiplied by the imaginary unit, form a basis for the group, which in turn exponentiate to the relevant Special Unitary Group . The algebra for the Pauli matrices involves a commutator and an anticommutator

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 (2)

Assign now  and  to two Pauli matrices, for instance

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 (3)
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 (4)

Next, to extract the coefficient of  from

to solve it for  we note that each term has a factor  multiplying a sum, so we only need to take into account the first  terms (sums) and in each sum replace  by the corresponding . For example, given to compute  we only need to compute these first three terms:

then solving for  one gets .

Also, since to compute  we only need the coefficient of , it is not necessary to compute all the terms of each multiple-sum. One way of restricting the multiple-sums to only one power of  consists of using multi-index summation, available in the Physics package (see Physics:-Library:-Add ). For that purpose, redefine sum to extend its functionality with multi-index summation

 >
 (5)

Now we can represent the same computation of  without multiple sums and without computing unnecessary terms as

Finally, we need a computational representation for the repeated commutator bracket

One way of representing this commutator bracket operation is defining a procedure, say F, with a cache to avoid recomputing lower order nested commutators, as follows

 >
 (6)
 >

For example,

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 (7)
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 (8)
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 (9)

We can set now the value of

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 (10)

and enter the formula that involves only multi-index summation

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 (11)

from where we compute  by solving for it the coefficient of , and since due to the mulit-index summation this expression already contains  as a factor,

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 (12)

In order to generalize the formula for H for higher powers of , the right-hand side of the multi-index summation limit can be expressed in terms of an abstract N, and H transformed into a mapping:

 >
 (13)

Now we have

 >
 (14)
 >
 (15)

The following is already equal to (11)

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 (16)

In this way, we can reproduce the results published in the literature for the coefficients of Zassenhaus formula up to  by adding two more multi-index sums to (13). Unassign  first

 >
 >

We compute now up to  in one go

 >
 (17)

The nested-commutator expression solved in the last step for  is

 >
 (18)

With everything understood, we want now to extend these results generalizing them into an approach to compute an arbitrarily large coefficient , then use that generalization to compute all the Zassenhaus coefficients up to . To type the formula for H for higher powers of  is however prone to typographical mistakes. The following is a program, using the Maple programming language , that produces these formulas for an arbitrary integer power of :

Formula := proc(A, B, C, Q)

This Formula program uses a sequence of summation indices with as much indices as the order of the coefficient  we want to compute, in this case we need 10 of them

 >
 (19)

To avoid interference of the results computed in the loop (17), unassign  again

 >

Now the formulas typed by hand, used lines above to compute each of ,  and , are respectively constructed by the computer

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 (20)
 >
 (21)
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 (22)

Construct then the formula for  and make it be a mapping with respect to N, as done for  after (16)

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 (23)

Compute now the coefficients of the Zassenhaus formula up to  all in one go

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 (24)

Notes: with the material above you can compute higher order values of . For that you need:

 1 Unassign  as done above in two opportunities, to avoid interference of the results just computed.
 2 Indicate more summation indices in the sequence  in (19), as many as the maximum value of n in .
 3 Have in mind that the growth in size and complexity is significant, with each  taking significantly more time than the computation of all the previous ones.
 4 Re-execute the input line (23) and the loop (24).
 >