Download Explicit solution of the continuous Baker-Campbell

ANNALS
OF PHYSICS:
51,
187-200 (1969)
Explicit Solution of the Continuous Baker-Campbell-Hausdorff
Problem and a New Expression for the Phase Operator
I. BIALYNICKI-BIRULA,
B. MIELNIK,
AND J. PLEBA~~SKI
Institute of Theoretical Physics, Warsaw University, Warsaw, Poland
An explicit formula for an arbitrary function of the evolution operator is derived.
With its use, the continuous analog of the Baker-Campbell-Hausdorff
problem is solved.
The application of this result to the quantum theory of scattering leads to a new closed
expression for the phase shifts in every order of perturbation theory.
I. INTRODUCTION
In many branches of physics and mathematics we are led to study the evolution
equation
(1.1)
The purpose of this paper is the derivation of a new representation for an
arbitrary function of E, the evolution 0perator.l Using this representation for the
function Sz = In E, we obtain the solution of the continuous analog of the
Baker-Campbell-Hausdorff
problem (I), (2) in a closed form.
We discuss also briefly in this paper one of the most interesting applications
of our result, namely, to the scattering theory in quantum mechanics, and we
derive an explicit formula for the phase shifts in every order of perturbation
theory.
The evolution equation (1.1) and the initial condition for the evolution operator,
J%l 9 44 = 1,
(1.2)
are equivalent to the following integral equation
1 We shall call traditionally all objects like A(t), E(r, &,) etc. the operators even though they
need not be defined as operators in a vector space. All we shall need is that they form an associative algebra.
187
188
BIALYNICKI-BIRULA,
MIELNIK,
AND
PLEBAfiSKI
The formal iterative solution2 of this equation has the form
where
The infinite series (1.4) is often written in the form of Dyson’s time-ordered
exponential (3)
W, to) = T ew (/la 4 4t3).
(1.6)
This representation of the evolution operator has been successfully used in
quantum theories, especially in quantum electrodynamics, to obtain approximate
expressions for the transition amplitudes.
The exponential representation (1.6) of the evolution operator is generalized
in this paper to an arbitrary functionf(E - 1) of the evolution operator. Assuming,
without any real loss of generality, thatf(0) = 1, we derive the following formula
forf(E - I),
f@ - 1)= Fexp(j:, 4 A(G),
where the ordering operation F depends on the function f and is a generalization
of the time-ordering of Dyson.
In Section II we derive, with the use of a resolvent operator, a simple closed
formula for f(E - l), called by us the canonical representation. This canonical
representation serves as an intermediate step in the derivation of (1.7), but has
also some direct applications.
In Section III we introduce the concept of the F-ordering operation and we
transform the canonical representation to the form (1.7). We study also in some
detail the ordering operation L connected with the logarithm of the evolution
operator.
In Section IV we apply our formulae for the logarithm of the evolution operator
to the quantum theory of scattering. We discuss some advantages of using this
a Problems of existence and covergence will not be investigated here. The formulae appearing
in this paper should be understood as purely algebraic relations in the spirit of formal power
series approach.
CONTINUOUS BAKER-HAUSDORFF
FORMULA
189
new representation for the phase operator in the scattering theory and we derive
an explicit integral formula for the phaseshift in every order of perturbation theory.
We restrict ourselves in this paper to an application in the scattering theory,
but we think that our results can be also applied in other fields of physics and
mathematics. To give just a few examples we may mention here possible applications to statistical physics (the Liouville equation, the master equation, the Bloch
equation, the Boltzmann equation), to group theory (construction of group
elements from generators) and to systemsof ordinary differential equations.
II. THE CANONICAL
REPRESENTATION
We show in this section, that every function f of the evolution operator, which
is analytic at z = 0, can be representedin the following canonical form
and
0, = o,(t, ,..., td = knwl + en-l,n--2+ -
+ 4,.
(2.3)
Function 0, takes on values 0, l,..., II - 1 and we set 0, = 0 consistently.
The canonical form off(T) is most easily found with the use of the resolvent
operator R,
where h is a complex parameter and
Ekl = E(tk - tL) = 2e(t, - tl) - 1.
(2.5)
The resolvent operator R reduces to T when x is set equal to 1.
R(t, to ; A)[,=, = +T(t, to).
(2.6)
3 We shall drop the index n and write 0, instead of 8,) whenever it will not lead to a confusion.
190
BIALYNICKI-BIRULA,
MIELNIK,
AND
PLEBAtiSKI
The resolvent operator can be brought to the canonical form with the use of the
following identity
; (%.,-1 + a *** ($1 + A) = & (A + l)@ (h - I)“-e-1.
(2.7)
The operator R obeys the following simple differential equation.
;
R(t, to ; h) = R2(r, t,, ; h).
w9
TO prove this we write R in a symbolic form,
Using the dot to denote the product of two operators, which contain independent
integrations with respect to tn .. . tI, and tkwl .. . tl , we can write
+~jA(E+X)A(E+X)A.~jA+..‘.
(2.10)
The right-hand side in this equation is easily seen to be the formal product of
two series (2.9).
The explicit solution of Eq. (2.8) is
R(t, to ; h) =
%(tP to)
1 - AR&, to) ’
(2.11)
where
Rdt, to) = R(t, t,, ; AL,
.
(2.12)
It follows from (2.1 I), that all powers of R can be expressed in terms of its
derivatives
Rn+l - 1 a”R
n! ahn’
Some additional
properties of R are discussed in the Appendix.
(2.13)
CONTINUOUS
BAKER-HAUSDORFF
191
FORMULA
Formula (2.11) can be now used to evaluate coefficient functions&(@). To this
end let us consider the contour integral representation of an analytic function
OfE1:
(2.14)
On account of (2.11) and (2.6) the evolution operator can be represented as
and the integral (2.14) can be rewritten in the form
(2.16)
The second term can be expressed in a compact form in terms of R, and the
following formula for f(E - 1) results:
f (E -
1) = f, + &
f dz 9
2R(t, t,, ; 1 + 2/z).
(2.17)
In order to obtain the representation (2.1) off(E - I), we introduce the expansion
(2.4) of R into (2.17) and make use of the identity
~(%~-l
+ 4 +*- cc21+ qh 1+2,2 = A(1
+ z)?
(2.18)
The integration over z in (2.2) can be carried out effectively in all cases of
physical interest. Several important examples are discussed below.
(a)
The Cayley transform C of the evolution operator:
c=
f@)
= &
E---l
-=->
E-l-1
T
2-l-T
$ g (1 + z)” &
(2.19)
= (-1)+-B--l
&.
(2.20)
With the use of the identity
(-l)n-e-1
we can transform
Schwinger (4),
= E,,,-1E,-1,12-2 *** E21,
the canonical
C = ; s:, dt, A(h) + nt2&
representation
(2.21)
of C to the form given by
s:, dt, *-- s” dtl c,z,sl ... c21A(t,J ... A&).
to
(2.22)
We notice also, that C coincides with the initial value R, of the generating operator.
192
BIALYNICKI-BIRULA,
(b)
MIELNIK,
AND
PLEBAIbKI
Arbitrary power ED of the evolution operator (JJmay be real or complex):
Ep = (1 + T)“,
(2.23)
(1 + zY+,’ = ; (0 + p)(O + p - 1) **a (0 + p - n + 1),
(2.24)
ED = 1 + nzl $ i:, dt, .*a it
to
dt,(O + p) *.* (0 + p - n - 1) A@,) -*- A&).
(2.25)
Known expansions for E and E-l are obtained from (2.26) with the use of the
identities
$(@
+ 1) *-- (0 - 12+ 2) = BQ&
$0
- 1) ‘.. (0 - n) = (-l)n
(c) The logarithm
Q of the evolution
= 8,&
ss,O = (-l)n
(2.26)
e,, a** 8,-r,, .
(2.27)
operator:
D = ln(1 + T),
E = eR,
(1 + z)@In(1 + z) = (-l>n-@-I
m=&+l
*** e,, ,
(2.28)
$ O!(n - 0 - l)!.
(2.29)
@!(n - 0 - I)! A(t,J .a. A@,).
(2.30)
The canonical representation of 1;2has the form
Q = $$
j:odh
*-* jt dt,(--l)+@-l
to
In the next section we study the properties of 52 in some detail.
III.
F-ORDERING
OPERATIONS
Since only the symmetric part of the integrand contributes to the n-fold integral,
we can rewrite the canonical representation (2.1) of f(E - 1) in the following
symmetrized form,
f(E -
1) = h + fjl;
f/n
-*- f.dt,
F(A(t,J
..a A(Q),
(3.1)
(3.2)
CONTINUOUS
BAKER-HAUSDORFF
193
FORMULA
and the sum is extended over all n ! permutations rr of the indices in ,..., il . We call
FtAn ... A,) the F-ordered product of operators A, ,..., A, . The simplest examples
of F-ordered products are well known T-ordered (chronological) and Bordered
(antichronological) products.
With the use of the same convention, that is adopted for the T-ordered products,
namely, that the ordering operation is to be performed before the integrations,
we can write (3.1) in the form
f(E - 1) = h - 1 + Few (j:,dh A(h)).
(3.3)
This formula reduces to (1.7) when, as we had assumed before,f, = 1.
Due to the symmetry of F(A(t,J *.a A(Q) with respect to tn ,..., t, , we can
reduce every integration in (3.1) to the integration over the simplex defined by
the inequalities
t, > tnel > *** > t, ;
(3.4)
f(E
- 1) = So + nfl jio dt, *** j;;h
Ft.Wn> a-* Aft,)).
(3.5)
When the conditions (3.4) are satisfied, @(t.%,,..., tc,) in (3.2) becomes a function
of the permutation rr only. We shall denote this function by O(r). Its value is
equal to the total number of chronologically ordered neighbors (i.e., tik > ti,-,)
in the permutation rr,
@(?T)
=
@(fin
)...,
ti,)
tn
>
tn-1
>
*-’
t,
.
(3.6)
We devote the rest of this section to the study of one important ordering
operation: that for the logarithm of E. We call the expression (3.2) in this case
the L-ordered product and we can express it for t, > *** > t, in the form
L(A(tJ
a-. A(&))
= $ c (-l)+@(“-l
* n
[e(r)]!
[n - @(z-) - l]! A(&,) -.. A(ti,).
(3.7)
For n = 1, 2, 3, 4, 5 we obtain
(3.8a)
(3.8b)
(3.8~)
(3.8d)
595/51/I-13
194
BIALYNICKI-BIRULA,
MIELNIK,
AND
PLEBArjSKI
+ %42A*A.¶A5A,l + ~A,AlA3A,A4~
+ bGbu2A31
+ ~~44,&444
+ 2{A,A4A,A,A,l
+ 3{AIA,A,A,A,l)>
where Ai = A(&) and i } denotes the multiple
MA-,
(3.8e)
commutator,
*.- 4 = [A,,, [An-1,..., [A,, 41 -**Il.
(3.9)
Integrating L(A(tJ *a* A@,)) according to the formula (3.5) we obtain the nth order
term in the expansion of Q. Lower order terms of this expansion have been
calculated in the literature before by iterative methods. Recently, Wilcox (5) carried
out this calculation up to n = 4.
It can be generally proved, that in every order of perturbation theory D is a
linear combination of integrated multiple commutators. This follows either from
the Friedrichs theorem (2) or more directly from the differential equation (A.7)
for Q derived in the Appendix of this paper. Knowing this multiple commutator
structure of 52, we can transform the r.h.s. in (3.7), with the use of the DynkinSpecht-Weaver theorem (2), to the form
L(A(t,)
a.. A@,)) = ;
c (- 1)R- e(n)-1 [63(n)]! [n - S(r) - l]! ; {A(&>
*R
*** A(&J}.
(W
It is actually this form of L that we used to evaluate formulae (3.8).
IV. PHASE OPERATOR
AND
PHASE SHIFTS
In this section we apply our results to the quantum scattering theory and we
derive an explicit integral expression for the phase shift in an arbitrary order
of perturbation theory. Our representation of the logarithm of the evolution
operator is expected to be especially useful in this case since all approximate
expressions for the phase operator obtained in perturbation theory would lead
to unitary evolution operators. No violation of unitarity, often found in ordinary
perturbation theory will appear in this method.
For simplicity, we restrict ourselves in this paper to the nonrelativistic quantum
mechanics. Applications to relativistic quantum mechanics and field theory will
be presented in a separate publication.
CONTINUOUS
BAKER-HAUSDORFF
195
FORMULA
Let us consider scattering of a particle with the mass p by the static potential I’.
The unitary evolution operator U(t, t) in the Dirac representation obeys the
equation
$
U(t, to) = -iVdt)
WC &I>,
(4.1)
where
J/,(t) = pot T/e-“w
(4.2)
and
Ho = @12/A.
(4.3)
With the use of the canonical representation (2.1) we can express any function f
of the S operator, S = U(co, - co), in the form
X (1 + ze,,,-,) ... (1 + zf&) eiHotnVe-i*o’t”-tn-l’V
*a* VemiHo’l. (4.4)
In the formula for the matrix element of f(S - l), in any representation in which
H,, is diagonal, all ti integrations can be carried out leaving us with the following
formula
(E’, 01’I (f (S - 1) - fo> I E, a>
(-1)”
= 6(E - E’) f
$ d[ f(l/LJ(E’,
a’ I VG,(E)V
a*. VG,(E)V
+
I E,
9Z=l
=
--SW
-
E')
f
4fU/W',
a'
I W
G,(E)V)-l
a>,
I E, a>
(4.5)
where
G,(E) = P H
0
and the integration in the <-plane is to be carried out along the closed contour
obtained by the inversion: 5 = l/z from the contour defined before in the complex
z-plane.
In the simplest case of a spherically symmetric potential we can use the complete
set of spherical waves to derive an explicit integral formula for the matrix
element (4.5).
We introduce the following notation
VdE,E’) = <E,1,m I V I E’, I, m>,
g,(E - E’) = P &
+ im(l + 25) S(E - E’),
(4.7)
(4.8)
196
BIALYNICKI-BIRULA,
MIELNIK,
AND
and use the completeness and orthonormality
PLEBArjSKI
of the set 1E, I, m),
~~~IE,I,m)dE(E,I,m/
(4.9a)
= 1,
(E, I, m 1E’, I’, ml> = S,,J,,4(E
(4.9b)
- E’)
and the relations
V 1E, I, m> = SW dE’ V,(E, E’) 1E’, I, m>,
(4.10)
0
(4.11)
G,tE’) I E, 6 m> = gdE - E’) I E, 4 m>.
Wavefunctions
(x 1E, 1, m) are well known spherical waves
<x I E, 4 m> = (+)l”
(4.12)
Jl+l12tkr) Y,“(y, 4,
where
(4.13)
k = (2t~E)‘/~,
so that the reduced matrix element
of V(r) and Bessel functions,
V,(E, E’) can be expressed as an integral
VdE, E’) = P jm dr rJz+,,,tkr)
V(r) J~+112Wr)-
(4.14)
0
With the help of Eqs. (4.7)-(4.11), we can transform the expression (4.5) to the
form
<E’, I’, m’ I (W
- 1) - .A,) I 4 6 m>
= 6,,&,4(E
- E’) 2 (-1)”
$ d5 f(1/5) j-y dE, *.. j.; dEnml
V&=1
* V,tE,
El)
gc&
- E)
VdE2
, Ed *** gd&-1 - E> VdL-1 , E).
The following integral relation between Bessel and Neumann
(4.15)
functions (6)
gdr, r’; E, 0 ; s* dE, Jt+l,2tk,r) g& - 9 J~+&~r’~
0
= - +Kr - r’> JJl+l,2(kr) Jl+&r’) + 4r’ - r) J~+dkr) ~~+dW
(4.16)
- W + 25) J1+1,2W)Jr+I,2W)l
enables us to express every term in (4.15) as an integral over 5 and r, ,..., r,, only.
CONTINUOUS
BAKER-HAUSDORFF
FORMULA
197
In order to illustrate this procedure, we use now (4.15) to evaluate the phase
shifts. In agreement with the standard definition, we introduce the phase shifts &(k)
through the formulae:
@ = 4 In S,
(E’, I’, m’ 1Q, 1E, Z, mj = 6,,4,,%3(E
From Eqs. (4.15)-(4.18)
(4.17)
- E’) 6,(k).
we obtain
. gdr, , r2 ; E, 0 VP,> ..* g&n-l , rn ; E, 0 W-d J~+l,2(krn).
It follows
(4.18)
(4.19)
from the relation
(4.20)
that all imaginary terms disappear from (4.19) so that 6,(k) is real, as it was to be
expected.
First two terms of the series (4.19) are
6i1’(k) = -np
jm drr(Jz+,,,(kr))2
V(r),
(4.21a)
0
@“‘W
=4~)~jmdr2r2Jz+l,2(kr2)
V(r2)
[N,+l,2(kr,)
0
- f’ d~~~~J~+~~2(W
WI> Jt+112(kr1)+ Jt+l,2(kr2)
0
* jm drlrlN~+l~2(W W-4 J,+l,2(krl)].
T2
(4.21b)
The same result is, of course, obtained by solving by iteration the integral equation
for the Jost function (7) or by any other iterative method. These methods give,
however, an algorithm only, whereas we have an explicit formula in every order
of perturbation theory.
To close this section, we apply our formula (4.19) to the scattering by the
“surface”
potential V(r) = Vo6(r - r). In that case all integrations
over ri’s
can be immediately carried out. The summation over n and the integration with
198
BIALYNICKI-BIRULA,
MIELNIK,
AND
PLEBAhKI
respect to 5 can also be easily performed. The final expression for the phase shift is:
in agreement with the result obtained from the Jost function.
APPENDIX
We derive here several properties of the resolvent operator R.
First, we notice, that the name resolvent is justified because R obeys the Hilbert
equation for the resolvent operators
R(t, to ; A) - W, to ; U = (4 - U R(t, fo ; 4) Rk to ; U.
(A-1)
This equation follows directly from the expression (2.4) for R.
Next, we derive the differential equation obeyed by R in the variable t.
Since
E(t, 4,) = 1 + Wt, 4, ; I),
64.2)
we can obtain from the Hilbert equation (A.l) an expression for R in terms of E,
R(t, to ; h) =
E(t, to) - 1
1 + A + (1 - A) W,
to) ’
After differentiating both sides of this equality with respect to t and using the
evolution equation, we obtain
3R
1
at=
21+)I+(I-~)EAEl+X+(I-~)E*
1
(A.4)
With the use of (A.3) we can rewrite this equation as a differential equation for R,
d& R(t, to ; X) = +[l - (1 - X) R(t, t, ; X)1 A(t)[l
+ (1 + A) R(t, t, ;
aI.
(A.5)
In three special cases, for h = fl and 0, Eq. (A.5) becomes the differential
equation for the evolution operator, for its inverse and for its Cayley transform.
aE/at = AE,
iYE-l/at = -E-IA,
aqat = +(i - c) A(I + c).
(A.6a)
(A.6b)
(A.W
CONTINUOUS
BAKER-HAUSDORFF
Finally, we derive, with the use of (A,5), the following
in the variable t for Q.
aJ-2
_ = A + -f 1
at
199
FORMULA
differential
[Q )..., [fi, A] -*- 1,
n=1
equation
(A.71
.
where B, are the Bernoulli numbers. This equation is most easily derived from the
compact formula for L?,
Q(t,
to)
= j1
dAR(t,
t,
; A),
68)
-1
which is obtained by an elementary integration
from the relation
1
R =
(A-9)
cth Ii’/2 - h *
When the canonical representation for the resolvent operator is substituted
into (A.8) and the integration over h is performed, our previous formula (2.30)
for 52 results.
We integrate now (A.4) with respect to X, use (A.8) and change the integration
variable X into p = 2 Arth h.
In the resulting equation for X2/i%,
aa
at=
1
1
A
sm
mmdP 1 + exp(Q + p)
1 + exp(-J2 - CL) ’
(A.10)
we use twice the integral representation
-= 1
and perform the integration
takes on the form
aS2
-=-
mdx
s -,mGe
(ix-l/P)Y
(A.11)
over ~1and one integration
over x. Eq. (A.10) then
1 + eV
rr m dx
at
(is-1/2)s2
/&iZ-l/2)"
2 s -,Ch2?TXe
In order to obtain Eq. (A.7), we use the multiple
product exp(zQ) A exp(-zQ), the formula
rr
T
m dx
s -,zze
(ix-1iz)t
--_
commutator
t
et - 1
(A.12)
expansion of the
(A.13)
200
BIALYNICKI-BIRULA,
and the definition
of the Bernoulli numbers B,, ,
MIELNIK,
AND
PLEBAhKI
(A. 14)
Differential equation
different method.
(A.7) has been obtained
by Magnus (I) by a completely
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A. KARRASS, AND D. SOLITAR, “Combinatorial
Group Theory,” Chap. 5.
Interscience, New York, 1966.
F. J. DYSON, Phys. Rev. 75, 486 (1949).
J. SCHWINGER,
Phys. Rev. 74, 1439 (1948).
R. M. WILCOX, J. Math. Phys. 8, 962 (1967).
G. N. WATSON, “A Treatise on the Theory of Bessel Functions,” Chap. 13. University Press,
Cambridge, 1922.
V. DE ALFARO AND T. REGGE, “Potential Scattering,” Chap. 4, North Holland, Amsterdam,
1965.
1. W. MAGNUS,
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