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Long time tails in decay theory
Volume 90A, number 5
PHYSICS LETTERS
12 July 1982
LONG TIME TAILS IN DECAY THEORY M.N. HACK 1616 East 50th Place, Chicago, IL 60615, USA Received 29 March 1982
The fact that quantum mechanical decay is necessarily slower than exponential at late times is established from first principles by a new method. The Paley—Wiener condition is not used, and no restriction is required on the state vector. An application is made to decay of classical correlations.
The quantum theory of decay, although dating al. most to the inception of quantum mechanics, continues to attract interest. It leads to the exponential law of decay, which applies to a wide variety of decay phenomena with a wide range of lifetimes of the decaying states and particles. As is well known, the exponential law can only be attained in quantum theory as an approximation, though in fact the theory predicts *1 that the approxi. mation usually will hold to such a high degree that no serious effort to look for departures from exponential decay experimentallyhas ever been considered. In applications one shows that the exponential law is valid in a broad middle range of values of time well encompassing the lifetime of the particular process, but at early and late times the exponential law necessarily breaks down according to quantum theory. Up to the present, the proof of this statement for large times [1] has relied on an appeal to the celebrat. ed theorem of Paley and Wiener [2]. Useful though it is, the proof based on the Paley—Wiener theorem raises the question whether the reliance on the Paley—Wiener condition can be avoided, and in the course of a recent study [3] it became clear that a proof from first principles based directly on analyticity should be possible. It is the purpose of this letter to present this proof. The necessarily long-tailed character of quantum mech. anical decay traces ultimately to the fact that although ~ Strong interaction resonances with widths comparable their central energies axe a possible exception. 220
an energy distribution with support bounded below can be infinitely smooth, i.e., differentiable to all orders everywhere, it nevertheless must have at least one singularity. In the following H denotes a seif-adjoint hamiltonian and t,(i an arbitrary state vector in Hilbert space. Units are chosen so that ~i = 1. The amplitude for finding the system, initially at a time t = 0 in the state ~i in the same state at time t ~ 0, is given by A’t’ =“ ‘~O~ ~,Ljtt”= ~1i e—~t,1, ‘ ‘
‘~“
r
=
“
‘ “
‘
e—Wt d’E~~ /
J
‘~
(1)
W~
—
where {E(u)} is the spectral family associated to H. We impose the condition normally applied in quanturn theory that the operator His lower semibounded, i.e., that the energy spectrum u(H)is bounded below. Letting m = mm u(H), we have (E(u)~,ti,~Li)=0foru
(2)
We shall prove that A(t) cannot decay exponentially fast as t —* °o. Suppose the contrary, Al \
.<
~
_~
c
ort
where C, a, and T are positive constants. We defme A(t) for t < 0 also by eq. (1), so that
to
1
4
A~t A~—t and —
0 03l-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 90A, number S A(t)l T
PHYSICS LETTERS -
(5)
A(t) is continuous by eq. (1) and the dominated convergence theorem. Since A(t) is therefore integrable, it follows by a well-known theorem on positive definite functions that the Fourier transform h(u) = ~
f e~A(t)dt
f e~’~h(u) clii
Khintchine representation. However in that case, of course, there is no basis to conclude vanishing spectral density on a convergent sequence of distinct points. We therefore have the result: A correlation function does not decay exponentially fast as t -+°° if the corresponding distribution function does not have a density which is analytic everywhere on the real line.
(6)
Note added in proof. The conditions for applica-
(7)
bility of the Paley—Wiener theorem can also be seen by the present method. Namely, by continuity and eq. (5),A(t) is square integrable and therefore, as a consequence of the Plancherel theorem, so is the Fourier transform h(u) which vanishes on a half-line. Alterna-
is integrable, and therefore A(t)
12 July 1982
by the Fourier inversion theorem for a continuous integrable function with an integrable Fourier transform. By (I) and (7) h(u) = d(E(u) i~i~ili)Idu (8) by the uniqueness theorem of Fourier—Stieltjes transforms *2 By virtue of the continuity of A(t) and eq. (5), a function h(z) of the complex variable z = u + iv in the strip urn zI
tively one could assume square integrabifity of A(t) instead of (3) (otherwise A(t) could not fall off exponentially fast) and calculate by the inversion and Plancherel theorems that (E(u) i~, ,ti) has a square integrable density. Thus, first of all at continuity points, and hence everywhere since the integral off(u) is a continuous function of its limits, (E(x)~Ji,~Ji)— (E(a)~/i,i/i) =
lim ~
f [(eixt
—
eiat)Iit]A(t) dt
—n =
urn
f f~(u)du f f(u) du, =
n-~
where f(u)
=
l.i.m. fn(u),
=
f~(u)
i
f
n
eiut A(t)
dt.
fl-+ 00
J h(u)du =A(0)=(~i,~~)> 0.
(9)
Hence the assumption (3) is untenable, and this completes the proof. The above argument applies to continuous stationary classical correlations as well, with the spectral measure furnished by the distribution function of the *2
*3
Finiteness of the measures is a consequence of f_00 d(E(u)~i,~‘)= (i’, iii) and (sinceh is in Li)f~ 00Ih(u)I x du <00. Alternatively, eq. (8) follows by the inversion formula for integrable characteristic functions. Or alternatively, by dominated convergence or by the Fubini and Morera theorems, for example.
(Li.msalimit inmean,i.e,in L~) References
-
[1] L.A. Khalfin, Soy. Phys. JETP 6 (1958) 1053. [2] R.E. Paley and N. Wiener, Fourier transforms in the cornplex domain (American Mathematical Society, New York, 1934) Theorem XII. [3] M.N. Hack, Phys. Left. 88A (1982) 228. [4] E.C. Titchmarsh, The theory of functions (Oxford University Press, London, 1939) P. 100. [5] R.E. Paley and N. Wiener, Fourier transforms in the cornplex domain (American Mathematical Society, New York, 1934) p. 3.
221
PHYSICS LETTERS
12 July 1982
LONG TIME TAILS IN DECAY THEORY M.N. HACK 1616 East 50th Place, Chicago, IL 60615, USA Received 29 March 1982
The fact that quantum mechanical decay is necessarily slower than exponential at late times is established from first principles by a new method. The Paley—Wiener condition is not used, and no restriction is required on the state vector. An application is made to decay of classical correlations.
The quantum theory of decay, although dating al. most to the inception of quantum mechanics, continues to attract interest. It leads to the exponential law of decay, which applies to a wide variety of decay phenomena with a wide range of lifetimes of the decaying states and particles. As is well known, the exponential law can only be attained in quantum theory as an approximation, though in fact the theory predicts *1 that the approxi. mation usually will hold to such a high degree that no serious effort to look for departures from exponential decay experimentallyhas ever been considered. In applications one shows that the exponential law is valid in a broad middle range of values of time well encompassing the lifetime of the particular process, but at early and late times the exponential law necessarily breaks down according to quantum theory. Up to the present, the proof of this statement for large times [1] has relied on an appeal to the celebrat. ed theorem of Paley and Wiener [2]. Useful though it is, the proof based on the Paley—Wiener theorem raises the question whether the reliance on the Paley—Wiener condition can be avoided, and in the course of a recent study [3] it became clear that a proof from first principles based directly on analyticity should be possible. It is the purpose of this letter to present this proof. The necessarily long-tailed character of quantum mech. anical decay traces ultimately to the fact that although ~ Strong interaction resonances with widths comparable their central energies axe a possible exception. 220
an energy distribution with support bounded below can be infinitely smooth, i.e., differentiable to all orders everywhere, it nevertheless must have at least one singularity. In the following H denotes a seif-adjoint hamiltonian and t,(i an arbitrary state vector in Hilbert space. Units are chosen so that ~i = 1. The amplitude for finding the system, initially at a time t = 0 in the state ~i in the same state at time t ~ 0, is given by A’t’ =“ ‘~O~ ~,Ljtt”= ~1i e—~t,1, ‘ ‘
‘~“
r
=
“
‘ “
‘
e—Wt d’E~~ /
J
‘~
(1)
W~
—
where {E(u)} is the spectral family associated to H. We impose the condition normally applied in quanturn theory that the operator His lower semibounded, i.e., that the energy spectrum u(H)is bounded below. Letting m = mm u(H), we have (E(u)~,ti,~Li)=0foru
(2)
We shall prove that A(t) cannot decay exponentially fast as t —* °o. Suppose the contrary, Al \
.<
~
_~
c
ort
where C, a, and T are positive constants. We defme A(t) for t < 0 also by eq. (1), so that
to
1
4
A~t A~—t and —
0 03l-9163/82/0000—0000/$02.75 © 1982 North-Holland
Volume 90A, number S A(t)l
PHYSICS LETTERS -
(5)
A(t) is continuous by eq. (1) and the dominated convergence theorem. Since A(t) is therefore integrable, it follows by a well-known theorem on positive definite functions that the Fourier transform h(u) = ~
f e~A(t)dt
f e~’~h(u) clii
Khintchine representation. However in that case, of course, there is no basis to conclude vanishing spectral density on a convergent sequence of distinct points. We therefore have the result: A correlation function does not decay exponentially fast as t -+°° if the corresponding distribution function does not have a density which is analytic everywhere on the real line.
(6)
Note added in proof. The conditions for applica-
(7)
bility of the Paley—Wiener theorem can also be seen by the present method. Namely, by continuity and eq. (5),A(t) is square integrable and therefore, as a consequence of the Plancherel theorem, so is the Fourier transform h(u) which vanishes on a half-line. Alterna-
is integrable, and therefore A(t)
12 July 1982
by the Fourier inversion theorem for a continuous integrable function with an integrable Fourier transform. By (I) and (7) h(u) = d(E(u) i~i~ili)Idu (8) by the uniqueness theorem of Fourier—Stieltjes transforms *2 By virtue of the continuity of A(t) and eq. (5), a function h(z) of the complex variable z = u + iv in the strip urn zI
tively one could assume square integrabifity of A(t) instead of (3) (otherwise A(t) could not fall off exponentially fast) and calculate by the inversion and Plancherel theorems that (E(u) i~, ,ti) has a square integrable density. Thus, first of all at continuity points, and hence everywhere since the integral off(u) is a continuous function of its limits, (E(x)~Ji,~Ji)— (E(a)~/i,i/i) =
lim ~
f [(eixt
—
eiat)Iit]A(t) dt
—n =
urn
f f~(u)du f f(u) du, =
n-~
where f(u)
=
l.i.m. fn(u),
=
f~(u)
i
f
n
eiut A(t)
dt.
fl-+ 00
J h(u)du =A(0)=(~i,~~)> 0.
(9)
Hence the assumption (3) is untenable, and this completes the proof. The above argument applies to continuous stationary classical correlations as well, with the spectral measure furnished by the distribution function of the *2
*3
Finiteness of the measures is a consequence of f_00 d(E(u)~i,~‘)= (i’, iii) and (sinceh is in Li)f~ 00Ih(u)I x du <00. Alternatively, eq. (8) follows by the inversion formula for integrable characteristic functions. Or alternatively, by dominated convergence or by the Fubini and Morera theorems, for example.
(Li.msalimit inmean,i.e,in L~) References
-
[1] L.A. Khalfin, Soy. Phys. JETP 6 (1958) 1053. [2] R.E. Paley and N. Wiener, Fourier transforms in the cornplex domain (American Mathematical Society, New York, 1934) Theorem XII. [3] M.N. Hack, Phys. Left. 88A (1982) 228. [4] E.C. Titchmarsh, The theory of functions (Oxford University Press, London, 1939) P. 100. [5] R.E. Paley and N. Wiener, Fourier transforms in the cornplex domain (American Mathematical Society, New York, 1934) p. 3.
221