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Trotter product formula

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The following formula for the exponential of two not necessarily commuting operators:

$$\exp(A+B)=\lim_{n\to\infty}(\exp(A/n)exp(B/n))^n.\tag{a1}$$

The easiest case of this is to see this as a (formal) identity in the completion (with respect to the augmentation ideal) of the free associative algebra over $\mathbf Q$ in the variables $A$ and $B$, where both $A$ and $B$ are given degree $1$.

The case of (a1) where $A$ and $B$ are $(n\times n)$-matrices is due to S. Lie, and is simply known as the product formula for matrix exponentials.

In the form

$$\exp(it(A+B))=s-\lim_{n\to\infty}(\exp(itA/n)\exp(itB/n))^n,$$

which is important in theoretical physics, it holds, e.g., when $A$ and $B$ are self-adjoint operators on a separable Hilbert space such that $A+B$, defined on the intersection of the domains of $A$ and $B$, is essentially self-adjoint. And in the form

$$\exp(-t(A+B))=s-\lim_{n\to\infty}(\exp(-tA/n)\exp(-tB/n))^n$$

it holds (for positive $t$) if $A$ and $B$ are bounded from below.

Collectively these results (and several more variants) are also known as the Trotter–Kato theorem.

The Trotter product formula finds many applications in quantum theory, both in theoretical and in simulation studies (of quantum spin systems, e.g).

References

[a1] H. Trotter, "On the product of semigroups of operators" Proc. Amer. Math. Soc. , 10 (1959) pp. 545–551
[a2] T. Kato, "Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups" I. Gohberg (ed.) M. Kac (ed.) , Topics in functional analysis , Acad. Press (1978) pp. 185–195
[a3] B. Simon, "Functional integration and quantum mechanics" , Acad. Press (1979)
[a4] E.B. Davies, "One-parameter semigroups" , Acad. Press (1980)
How to Cite This Entry:
Trotter product formula. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Trotter_product_formula&oldid=32830