Reply to “Comment on ‘On the inconsistency of the
Bohm-Gadella theory with quantum mechanics’ ”
Rafael de la Madrid
Department of Physics, University of California at San Diego, La Jolla, CA 92093
E-mail: rafa@physics.ucsd.edu
Abstract. In this reply, we show that when we apply standard distribution theory to
the Lippmann-Schwinger equation, the resulting spaces of test functions would comply
with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov,
and of the theory of ultradistributions were wrong. As well, we point out several
differences between the “standard method” of constructing rigged Hilbert spaces in
quantum mechanics and the method used in Time Asymmetric Quantum Theory.
PACS numbers: 03.65.-w, 02.30.Hq
1. Introduction
The authors of [1] allege to have shown that the conclusions of [2] regarding the
inconsistency of Time Asymmetric Quantum Theory (TAQT) with quantum mechanics
are false. In this reply, we will show that the arguments of [1] are missing essential
aspects of [2], and that therefore the conclusions of [2] still stand.
The most important claims of [1] are the following:
1. There are many examples of TAQT, and the present author has inadvertently
constructed another one.
2. The flaws of the Quantum Arrow of Time (QAT) pointed out in [2] are actually not
flaws, because the original derivation of the QAT was misquoted from its source [3].
3. The crucial argument of [2] regarding the exponential blowup of the test functions
ϕ̂±(z) does not prevent ϕ̂±(z) from being of Hardy class.
As we shall see, all these claims do not stand close scrutiny. In order to show why,
in Sec. 2 we will outline the method to construct rigged Hilbert spaces in quantum
mechanics based on the theory of distributions [4]. We shall refer to this method as
the “standard method” and show that the resulting rigged Hilbert spaces are not of
Hardy class. We shall also explain the meaning of the exponential blowup of ϕ̂±(z) and
why it implies that the spaces of test functions are not of Hardy class. In Sec. 3, we
briefly outline the method to introduce rigged Hilbert spaces of Hardy class in TAQT
and compare such method with the “standard method.” It will then be apparent that
using the method of TAQT, one can introduce any arbitrary rigged Hilbert space for the
http://arxiv.org/abs/0704.1613v1
Gamow states. In order to address claim 2, we show (again) in Sec. 4 that no matter
how one introduces it, the Quantum Arrow of Time has little to do with the actual time
evolution of a quantum system. To address claim 3, in Sec. 5 we use classic results of
Paley and Wiener and of Gelfand and Shilov to show that the “standard method” of
dealing with the Lippmann-Schwinger equation leads to rigged Hilbert spaces that are
not of Hardy class. Section 7 concludes that the arguments of [2] still stand.
2. The “standard method”
In this section, we illustrate the main features of the “standard method” to construct
rigged Hilbert spaces in quantum mechanics [5]. Such “standard method” is based on
the theory of distributions [4]. For the sake of clarity, we shall use the spherical shell
potential of height V0,
V (~x) = V (r) =
0 0 < r < a
V0 a < r < b
0 b < r < ∞ .
(2.1)
For l = 0, the Hamiltonian acts as (we take ~2/2m = 1)
H = − d
+ V (r) . (2.2)
The regular solution is
χ(r;E) =
E r) 0 < r < a
J1(E)ei
E−V0 r + J2(E)e−i
E−V0 r a < r < b
J3(E)ei
E r + J4(E)e−i
E r b < r < ∞ .
(2.3)
The Jost functions and the S matrix are given by
J+(E) = −2iJ4(E) , J−(E) = 2iJ3(E) , (2.4)
S(E) =
J−(E)
J+(E)
. (2.5)
The solutions of the Lippmann-Schwinger equation can be written as
〈r|E±〉 ≡ χ±(r;E) =
χ(r;E)
J±(E)
. (2.6)
When V tends to zero, these eigensolutions tend to the “free” eigensolution:
〈r|E〉 ≡ χ0(r;E) =
E r) . (2.7)
These eigenfunctions are delta-normalized and therefore their associated unitary
operators,
(U±f)(E) =
dr χ±(r;E) f(r) ≡ f̂±(E) , E ≥ 0 , (2.8)
(U0f)(E) =
dr χ0(r;E) f(r) ≡ f̂0(E) , E ≥ 0 , (2.9)
transform from L2([0,∞), dr) onto L2([0,∞), dE).
The Lippmann-Schwinger and the “free” eigenfunctions can be analytically
continued from the scattering spectrum into the whole complex plane. We shall denote
such analytically continued eigenfunctions by χ±(r; z) and χ0(r; z). Whenever they
exist, the analytic continuations of (2.8) and (2.9) are denoted by
f̂±(z) =
dr χ±(r; z) f(r) , (2.10)
f̂0(z) =
dr χ0(r; z) f(r) , (2.11)
where here and in the following z belongs to a two-sheeted Riemann surface.
The resonant energies are given by the poles zn of the S matrix, and their associated
Gamow states are
u(r; zn) = Nn
J3(zn) sin(
zn r) 0 < r < a
J1(zn)
J3(zn)e
zn−V0 r +
J2(zn)
J3(zn)e
zn−V0 r a < r < b
zn r b < r < ∞ ,
(2.12)
where Nn is a normalization factor.
The theory of distributions [4] says that a test function ϕ(r) on which a distribution
d(r) acts is such that the following integral is finite:‡
〈ϕ|d〉 ≡
dr ϕ(r)d(r) < ∞ , (2.13)
where 〈ϕ|d〉 represents the action of the functional |d〉 on the test function ϕ. With some
variations, this is the “standard method” followed by [7–15] to introduce spaces of test
functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the
method followed by the present author runs (somewhat) parallel to [8], not to TAQT.
In order to use (2.13) to construct the rigged Hilbert spaces for the analytically
continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to
obtain the growth of χ±(r; z), χ0(r; z) and u(r; zn). Because the regular solution blows
up exponentially [16],
|χ(r; z)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r , (2.14)
the growth of the eigenfunctions (2.6), (2.7) and (2.12) blows up exponentially:
|χ±(r; z)| ≤ C 1
J±(z)
|z|1/4 r
1 + |z|1/2 r
z |r , (2.15)
‡ In quantum mechanics, we need to impose a few more requirements, but we will not need to go into
such details here.
|χ0(r; z)| ≤ C
|z|1/4 r
1 + |z|1/2 r
z |r , (2.16)
|u(r; zn)| ≤ Cn
|zn|1/2 r
1 + |zn|1/2 r
zn |r . (2.17)
When we plug this exponential blowup into the basic requirement (2.13) of the “standard
method,” we see that the test functions on which those distributions act must fall off at
least exponentially.
By using the Gelfand-Shilov theory of M an Ω functions [4], it was shown in [15]
that when a and b are positive real numbers satisfying
= 1 , (2.18)
and when ϕ+(r) is an infinitely differentiable function whose tails fall off like e−r
a/a, then
ϕ+(z) grows like e|Im(
z)|b/b in the infinite arc of the lower half-plane of the Riemann
surface:
If |ϕ+(r)| < Ce−
a as r → ∞, then |ϕ̂+(z)| ≤ Ce
b as |z| → ∞ . (2.19)
It was shown in [2] that when ϕ+(r) ∈ C∞0 , ϕ̂+(z) blows up exponentially in the infinite
arc of the lower half-plane of the Riemann surface:
If |ϕ+(r)| = 0 when r > A, then |ϕ̂+(z)| ≤ CeA|Im
z | as |z| → ∞ . (2.20)
From the above estimates, we concluded in [2] that the ϕ+’s obtained from the “standard
method” cannot be Hardy functions, since ϕ̂+(z) does not tend to zero as |z| tends to
infinity.
The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞
from estimates such as (2.19) or (2.20), and therefore they conclude that nothing
prevents ϕ̂+(z) from tending to zero and therefore from being Hardy functions. Their
conclusion is not true, because their argument does not take the nature of (2.19) and
(2.20) into account. After we explain the meaning of those estimates, it will be clear
why they prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface.
In order to understand what (2.19) and (2.20) mean, we start with the simple sine
function sin(
E r). When E ≥ 0, the sine function oscillates between +1 and −1:
| sin(
E r)| ≤ 1 , E ≥ 0 . (2.21)
As E tends to infinity, such oscillatory behavior remains, and in such limit the sine
function does not tend to zero. When we analytically continue the sine function,
z r) , (2.22)
the oscillations are bounded by
| sin(
z r)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r . (2.23)
Thus, as |z| tends to infinity, sin(
z r) oscillates wildly, and the magnitude of its
oscillation is tightly bounded by the exponential function. It is certain that as |z|
tends to infinity, sin(
z r) does not tend to zero, even though the function vanishes
z r = ±nπ, n = 0, 1, . . .
It just happens that the solutions of the Lippmann-Schwinger equation follow the
same pattern. When E is positive, the eigensolutions are oscillatory and bounded by
|χ±(r;E)| ≤ C 1
J±(E)
|E|1/4 r
1 + |E|1/2 r
. (2.24)
When the energy is complex, their oscillations get wild and are bounded by Eq. (2.15).§
Thus, the analytic continuations of the Lippmann-Schwinger eigenfunctions oscillate
wildly, and the magnitude of their oscillation is tightly bounded by an exponential
function (multiplied by factors that don’t cancel the exponential blowup when |z| → ∞).
Because in Eqs. (2.10) and (2.11) we are integrating over r, the exponentially-
bounded oscillations of χ±(r; z) get transmitted into ϕ̂±(z). The estimates (2.19) and
(2.20) bound the oscillation of the test functions of the “standard method,” except
for factors that don’t cancel the exponential blowup. It is the exponentially-bounded
oscillations of ϕ̂±(z) what prevent ϕ̂±(z) from tending to zero in any infinite arc of the
Riemann surface and therefore from being of Hardy class.
A somewhat simpler way to understand the above estimates is by looking at the
“free” incoming and outgoing wave functions ϕin and ϕout. Because in the energy
representation such wave functions are the same as the “in” and “out” wave functions,
ϕ̂in(E) = 〈E|ϕin〉 = 〈+E|ϕ+〉 = ϕ̂+(E) , (2.25)
ϕ̂out(E) = 〈E|ϕout〉 = 〈−E|ϕ−〉 = ϕ̂−(E) , (2.26)
in TAQT the analytic continuation of ϕ̂in(E) and ϕ̂out(E) are also of Hardy class. Since
ϕ̂in,out(z) =
z r)ϕin,out(r) , (2.27)
it is evident that the exponential blowup (2.23) of sin(
z r) will prevent ϕ̂in,out(z) from
tending to zero as |z| → ∞ in any half-plane of the Riemann surface. Thus, ϕ̂in,out(z)
are not of Hardy class, contrary to TAQT.
Strictly speaking, the bounds (2.19) and (2.20) are not the tightest ones. We should
include polynomial corrections, see Eq. (B.15) in [15], and the effect of
|z|1/4r
1+|z|1/2r
and 1J±(z)
to obtain the tightest bounds. We shall not obtain those corrections here, because they
do not cancel the exponential blowup at infinity, and because in this reply we shall use
instead other classic bounds, see Sec. 5.
Let us summarize this section. In standard quantum mechanics, once the
Lippmann-Schwinger equation is solved, the properties of ϕ̂±(z) are already determined
by Eqs. (2.10) and (2.11), and there is no room for any extra assumption on their
properties. This means, in particular, that the Hardy axiom cannot be simply assumed.
Rather, the Hardy axiom must be proved using Eqs. (2.10) and (2.11).‖ It simply
§ The points at which J±(z) = 0 do not affect the essence of the argument.
‖ This is what in [2] it was meant by the assertion that the Hardy axiom is not a matter of assumption
but a matter of proof.
happens that the “standard method” yields ϕ̂±(z) and ϕ̂in,out(z) that oscillate wildly.
Because these oscillations are bounded by exponential functions, ϕ̂±(z) and ϕ̂in,out(z) do
not tend to zero as |z| tends to infinity in any half-plane of the Riemann surface—hence
they are not of Hardy class.
3. TAQT vs. the “standard method”
In TAQT, one doesn’t solve the Lippmann-Schwinger equation in order to afterward
obtain the properties of ϕ̂±(z) using Eq. (2.10). Instead, one transforms into the energy
representation (using U± in our example) and then imposes the Hardy axiom. If H2±
denotes the spaces of Hardy functions from above (+) and below (−), S denotes the
Schwartz space, and Φ̃± denote their intersection restricted to the positive real line,
Φ̃± = H2± ∩ S|R+ , (3.1)
then the Hardy axiom states that the functions ϕ̂±(z) belong to Φ̃∓:
ϕ̂±(z) ∈ Φ̃∓ . (3.2)
This means that in the position representation, the Gamow states and the analytic
continuation of the Lippmann-Schwinger eigenfunctions act on the following spaces:
ΦBG∓ = U
± Φ̃∓ . (3.3)
It is obvious that the choices (3.2)-(3.3) are arbitrary. One may as well choose another
dense subset of L2([0,∞), dE) with different properties and obtain a different space of
test functions for the Gamow states. What is more, ΦBG± are different from the spaces
of test functions obtained through the “standard method,” because the functions ϕ̂±(z)
of the “standard method” are not of Hardy class.
The authors of [1] claim that the present author has inadvertently constructed
an example of TAQT. That such is not the case can be seen not only from the
differences between the “standard method” and the method used in TAQT to introduce
rigged Hilbert spaces, but also from the outcomes. For example, whereas in the
position representation the “standard method” calls for just one rigged Hilbert
space for the Gamow states and for the analytically continued Lippmann-Schwinger
eigenfunctions [15], TAQT uses two rigged Hilbert spaces
ΦBG± ⊂ L2([0,∞), dr) ⊂ Φ×BG± . (3.4)
One of the rigged Hilbert spaces is used for the “in” solutions and for the anti-resonant
states, whereas the other one is used for the “out” solutions and for the resonant states.
Another difference is that in TAQT, the solutions of the Lippmann-Schwinger equation
for scattering energies have a time asymmetric evolution [17], whereas the “standard
method” yields that such time evolution runs from t = −∞ to t = +∞, see [14].
Incidentally, this is an instance where TAQT differs not only mathematically but also
physically from standard quantum mechanics, because in standard scattering theory,
the time evolution of a scattering process goes from the asymptotically remote past
(t → −∞) to the asymptotically far future (t → +∞). This is not so in TAQT [17].
It seems hardly necessary to clarify what the present author means by “standard
quantum mechanics.” Standard quantum mechanics means the Schrödinger equation,
and standard scattering theory means the Lippmann-Schwinger equation. In standard
quantum mechanics, one assumes that these equations describe the physics and then
solves them. Because of the scattering and resonant spectra, their solutions lie within
rigged Hilbert spaces. The construction of such rigged Hilbert spaces follows by
application of the “standard method.” By contrast, TAQT simply assumes that the
solutions of the Schrödinger and the Lippmann-Schwinger equations comply with the
Hardy axiom, without ever showing that the actual solutions of those equations comply
with such axiom.
It was claimed in [2] that there is no example of TAQT. The authors of [1] dispute
such claim and assert that there are many examples. The present author disagrees
with their assertion, because assuming that for a large class of potentials the solutions
of the Lippmann-Schwinger equation comply with the Hardy axiom is not the same
as having an example where it is shown that the actual solutions of the Lippmann-
Schwinger equation comply with the Hardy axiom. In fact, to the best of the present
author’s knowledge, no advocate of TAQT has ever used Eq. (2.10) to discuss the
analytic properties of ϕ̂±(E) = 〈±E|ϕ±〉 in terms of the actual solutions χ±(r;E) of
the Lippmann-Schwinger equation.
The authors of [1] inadvertently acknowledge that there is no example of TAQT
when they say that they still need “to identify the form and properties” of the functions
of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging
that they don’t know whether the standard Gamow states defined in the position
representation are well defined as functionals acting on ΦBG±. If TAQT had an example,
it would be known.
4. The Quantum Arrow of Time (QAT)
Advocates of TAQT argue that their choice (3.3) is not arbitrary but rather is rooted
on a causality principle. Such causality principle is the “preparation-registration arrow
of time,” sometimes referred to as the “Quantum Arrow of Time” (QAT). For the “in”
states ϕ+, the causal statement of the QAT is written as
ϕ̃+(t) ≡
dE e−iEtϕ̂+(E) = 0 , for t > 0 . (4.1)
By one of the Paley-Wiener theorems, Eq. (4.1) is equivalent to assuming that ϕ̂+(E)
is of Hardy class from below. The corresponding causal statement for the “out” wave
functions ϕ− implies that ϕ− is of Hardy class from above. Hence, in TAQT, the
choice (3.3) is not arbitrary but a consequence of causality.
It was pointed out in [2] that the QAT is flawed. The argument was twofold. First,
it was pointed out that the original derivation [3] of Eq. (4.1) made use of the following
flawed assumption:
0 = 〈E|ϕin(t)〉 = 〈+E|ϕ+(t)〉 = e−iEtϕ̂+(E) , for all energies, (4.2)
which can happen only when ϕ+ and ϕin are identically 0. It was then pointed out that
even though one may simply assume the causal statement (4.1) and forget about how
it was derived, such causal statement says little about the actual time evolution of a
quantum system, because the quantum mechanical time evolution of ϕ+ is not given by
Eq. (4.1):
ϕ+(t) = e−iHtϕ+ 6= ϕ̃+(t) . (4.3)
To counter this argument, the authors of [1] claim that the derivation of the QAT
was misquoted from the original source [3], and that the flawed assumption (4.2) was
never used to derive the QAT (4.1). It seems therefore necessary to quote the original
derivation (see [3], page 2597):¶
“We are now in the position to give a mathematical formulation of the QAT: we
choose t = 0 to be the time before which all preparations of φin(t) are completed and
after which the registration of ψout(t) begins. This means that for t > 0 the energy
distribution of the preparation apparatus must vanish: 〈E, η|φin(t)〉 = 0 for all values
of the quantum numbers E and η (η are the additional quantum numbers which we
usually suppress). As the mathematical statement for ‘no preparations for t > 0’ we
therefore write (the slightly weaker condition)
dE 〈E|φin(t)〉 =
dE 〈+E|φ+(t)〉 =
dE 〈+E|e−iHt|φ+〉 (4.4)
dE 〈+E|φ+〉e−iEt ≡ F(t) for t > 0 . (4.5)
The readers can decide whether or not the flawed hypothesis (4.2) was used to derive
the QAT (4.5).
Nevertheless, it is actually not very relevant whether the authors of [3] used (4.2)
to derive (4.1). As pointed out in [2], and as mentioned above, even though one can
forget (4.2) and simply assume (4.1) as the causal condition to be satisfied by ϕ+, such
causal condition has little to do with the time evolution of a quantum system, see again
Eq. (4.3). In particular, as even the author of [6] has asserted, the t that appears in
Eq. (4.1) is not the same as the parametric time t that labels the evolution of a quantum
system.+ Thus, as far as standard quantum mechanics is concerned, the causal content
of the QAT is physically vacuous, and therefore, regardless of how one motivates it,
there is no physical justification for the choice (3.3).
5. TAQT vs. the “classic results”
In this section, we are going to compare the Hardy axiom of TAQT with some
classic results of Paley and Wiener, of Gelfand and Shilov and of the theory of
ultradistributions, which we shall collectively refer to as the “classic results.” More
¶ In this quote, φin, φ+, F(t) and Eq. (4.5) correspond, respectively, to ϕin, ϕ+, ϕ̃+(t) and Eq. (4.1).
+ All this shows that the new term TAQT is a misnomer. A better name is Bohm-Gadella theory,
because it was these two authors who proposed the theory and summarized it in [18].
precisely, we will see that the spaces of test functions ϕ̂± obtained by the “standard
method” would be of Hardy class only if the “classic results” were wrong.
The direct comparison with the “classic results” is more easily done in one
dimension, and therefore we shall use the example of the one-dimensional rectangular
barrier potential:
V (x) =
0 −∞ < x < a
V0 a < x < b
0 b < x < ∞ .
(5.1)
For this potential, the “in” and “out” eigensolutions are well known and can be found
for example in [12]. We shall denote them by χ±l,r(x;E), where the labels l,r denote
left and right incidence. When we analytically continue these eigenfunctions, or when
we consider the Gamow states for this potential, the “standard method” calls for test
functions ϕ±l,r(x) for which the following integrals are finite:
ϕ̂±l,r(z) =
dxχ±l,r(x; z)ϕ(x) . (5.2)
Just as in the example discussed in Sec. 2, the test functions ϕ(x) must at least fall off
faster than exponentials.
To further simplify the discussion, we need to recall that, because of Eqs. (2.25)
and (2.26), the Hardy axiom assumes that the “free” wave functions ϕ̂inl,r(E) and ϕ̂
l,r (E)
are also of Hardy class. These “free” functions are given by (hereafter, we just consider
ϕinl,r, since the analysis for ϕ
l,r is the same)
ϕ̂inl (E) =
dx e−ikx ϕin(x) , (5.3)
ϕ̂inr (E) =
dx eikx ϕin(x) , (5.4)
where k =
E is the wave number. The total wave function is given by the sum of left
and right components:
ϕ̂in(E) = ϕ̂inl (E) + ϕ̂
r (E) . (5.5)
It is simpler to work with k rather than with E and define
ϕ̂inl,r(k) ≡
2k ϕ̂inl,r(E) ; (5.6)
that is,
ϕ̂inl (k) =
dx e−ikx ϕin(x) , k ≥ 0 , (5.7)
ϕ̂inr (k) =
dx eikx ϕin(x) , k ≥ 0 . (5.8)
The “total” wave function in the wave-number representation, ϕ̂in(k) = ϕ̂inl (k)+ ϕ̂
r (k),
is thus the Fourier transform of ϕ(x),
ϕ̂in(k) =
dx e−ikx ϕin(x) , k ∈ R . (5.9)
Its analytic continuation will be denoted as
ϕ̂in(q) =
dx e−iqx ϕin(x) , q ∈ C . (5.10)
At this point, we are ready to introduce two classic theorems. The first one is due
to Paley and Wiener (see Theorem IX.11 in [19]):
Theorem 1 (Paley-Wiener). An entire analytic function ϕ̂(q) is the Fourier
transform of a C∞0 (R) function ϕ(x) with support in the segment {x | |x| < A} if,
and only if, for each N there is a CN so that
|ϕ̂(q)| ≤ CN e
A|Im(q)|
(1 + |q|)N
(5.11)
for all q ∈ C.
This theorem says that the Fourier transform of a C∞0 function is an analytic
function that grows exponentially, and that such exponential growth is mildly corrected
(but not canceled) by a polynomial falloff.
The second theorem we shall use is due to Gelfand and Shilov [4]. Before stating it,
we need some definitions. Let a and b denote two positive real numbers satisfying (2.18).
Let us define Φa,b as the set of all differentiable functions ϕ(x) (−∞ < x < ∞) satisfying
the inequalities
dnϕ(x)
∣∣∣∣ ≤ Cne
−α |x|
a (5.12)
with constants Cn and α > 0 which may depend on the function ϕ. Let us define the
space Φ̂a,b as the set of entire analytic functions ϕ̂(q), q = Re(q) + i Im(q), which satisfy
the inequalities
|qnϕ̂(q)| ≤ Cne+β
|Im(q)|b
b , (5.13)
where the constants Cn and β > 0 depend on the function ϕ. It is obvious that
the elements of Φa,b are functions that, together with their derivatives, decrease at
infinity faster than e−
a , whereas the elements of Φ̂a,b are analytic functions that grow
exponentially at infinity as e+
|Im(q)|b
b , except for a polynomial correction that doesn’t
cancel the exponential blowup.
Theorem 2 (Gelfand-Shilov). The space Φ̂a,b is the Fourier transform of Φa,b.
This theorem means that the smooth functions that fall off at infinity faster than
e−|x|
a/a are, in Fourier space, analytic functions that grow exponentially like e+|Im(q)|
The bounds (5.11) and (5.13) are to be understood in the same way as the
bounds (2.19) and (2.20). That is, the bounds (5.11) and (5.13) mean that ϕ̂(q) is
an oscillatory function that grows exponentially in the infinite arc of the q-plane, the
oscillation being tightly bounded by Eqs. (5.11) and (5.13) when ϕ(x) belongs to C∞0
and Φa,b, respectively. Note that after the addition of the corresponding polynomial
corrections, the bounds (2.19) and (2.20) are entirely analogous to the bounds (5.11)
and (5.13)—the operators U± are after all Fourier-like transforms [12].
Let us now apply the above theorems to the functions ϕin(x) obtained by the
“standard method.” In order for Eq. (5.10) to make sense, ϕin(x) must fall off faster
than exponentials. If we choose ϕin(x) to fall off like e−|x|
a/a, then the Gelfand-Shilov
theorem tells us that ϕ̂in(q) grows like e+|Im(q)|
b/b. Even when we impose that ϕin(x)
is C∞0 , which is already a very strict requirement, the Paley-Wiener theorem says that
ϕ̂in(q) grows exponentially. This means, in particular, that the ϕ̂in(q) do in general not
tend to zero in the infinite arc of the q-plane, because if they did, the Paley-Wiener
and the Gelfand-Shilov theorems would be wrong. Because of Eq. (5.6), ϕ̂in(z) does in
general not tend to zero as |z| tends to infinity in the lower half-plane of the second
sheet. Hence the space of ϕ̂in’s is not of Hardy class from below.
The space of ϕ̂+’s cannot be of Hardy class from below either, because if it were,
|z|→∞
ϕ̂+(z) = 0 , (5.14)
where the limit is taken in the lower half plane of the second sheet. By Eq. (2.25),
this implies that also the space of ϕ̂in’s would be of Hardy class and comply with this
limit, which we know is not possible due to the “classic results.” Thus, the “standard
method” yields spaces of test functions that do not comply with the Hardy axiom. This
is precisely what it was meant in [2] by the assertion that TAQT is inconsistent with
standard quantum mechanics.
To finish this section, we note that if we chose the test functions as in [8], then
we would be dealing with ultradistributions. In Fourier space, the test functions for
ultradistributions grow faster than any exponential as we follow the imaginary axis,
see [8] and references therein. Thus, if the “standard method” yielded spaces of Hardy
functions, that property of ultradistributions would be false.
6. Further remarks
The authors of [1] claim that it is inaccurate to state that the proponents of TAQT
dispense with asymptotic completeness. This statement should be compared with the
first quote in section 6 of [2].
The authors of [1] also claim that TAQT obtains the resonant states by solving
the Schrödinger equation subject to purely outgoing boundary conditions. This claim
should be compared with the second quote in section 6 of [2].
The authors of [1] also dispute the assertion of [2] that TAQT sometimes uses the
whole real line as though it coincided with the scattering spectrum of the Hamiltonian.
A glance at, for example, the QAT (4.1) seems to support such assertion.
7. Conclusions
In standard scattering theory, one assumes that the physics is described by the
Lippmann-Schwinger equation. When one solves such equation, one finds that its
solutions must be accommodated by a rigged Hilbert space, and that its time evolution
runs from t = −∞ till t = +∞ [14]. When one analytically continues the solutions
of the Lippmann-Schwinger equation, one finds that they must be accommodated by
one rigged Hilbert space, which also accommodates the resonant (Gamow) states. The
construction of such rigged Hilbert space is determined by standard distribution theory.
By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger
equations belong to two rigged Hilbert spaces of Hardy class. In TAQT, one never
explicitly solves the Lippmann-Schwinger equation for specific potentials in the position
representation. Instead, one assumes that its solutions satisfy the Hardy axiom. Unlike
in standard scattering theory, in TAQT the time evolution of the solutions of the
Lippmann-Schwinger equation does not run from t = −∞ till t = +∞.
By comparing the properties of the actual solutions of the Lippmann-Schwinger
equation with the Hardy axiom, we have seen that such actual solutions would comply
with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov,
and of the theory of ultradistributions were wrong. We have (again) stressed the fact
that the Quantum Arrow of Time, which is the justification for using the rigged Hilbert
spaces of Hardy class, has little to do with the time evolution of a quantum system. We
have stressed that using the method of TAQT to introduce rigged Hilbert spaces, we
could accommodate the Gamow states in a landscape of arbitrary rigged Hilbert spaces,
see also [5].
Our claim of inconsistency should not be taken as a claim that TAQT is
mathematically inconsistent or that TAQT doesn’t have a beautiful mathematical
structure. What the present author claims is that TAQT is not applicable in quantum
mechanics and is in fact a different theory.
To finish, we would like to mention that the “classic theorems” are not in conflict
with using Hardy functions in quantum mechanics. They are in conflict only with the
Hardy axiom. Thus, our results do not apply to other works that use Hardy functions
in a different way [20].
Acknowledgment
This research was supported by MEC and DOE.
References
[1] M. Gadella, S. Wickramasekara, “A Comment on ‘On the inconsistency of the Bohm-Gadella
theory with quantum mechanics,’ ” J. Phys. A (to be published).
[2] R. de la Madrid, J. Phys. A: Math. Gen. 39, 9255 (2006); quant-ph/0606186.
[3] A. Bohm, I. Antoniou, P. Kielanowski, J. Math. Phys. 36, 2593 (1995).
http://arxiv.org/abs/quant-ph/0606186
[4] I.M. Gelfand, G.E. Shilov Generalized Functions, Vol. 1-3, Academic Press, New York (1964).
I.M. Gelfand, N.Ya. Vilenkin Generalized Functions Vol. 4, Academic Press, New York (1964).
[5] R. de la Madrid, “Reply to [6],” submitted to J. Phys. A.
[6] H. Baumgartel, “Comment on [2],” submitted to J. Phys. A.
[7] G. Parravicini, V. Gorini, E.C.G. Sudarshan, J. Math. Phys. 21, 2208 (1980).
[8] C.G. Bollini, O. Civitarese, A.L. De Paoli, M.C. Rocca, Phys. Lett. B382, 205 (1996);
J. Math. Phys. 37, 4235 (1996).
[9] R. de la Madrid, A. Bohm, M. Gadella, Fortschr. Phys. 50, 185 (2002); quant-ph/0109154.
[10] R. de la Madrid, J. Phys. A: Math. Gen. 35, 319 (2002); quant-ph/0110165.
[11] R. de la Madrid, Int. J. Theo. Phys. 42, 2441 (2003); quant-ph/0210167.
[12] R. de la Madrid, J. Phys. A: Math. Gen. 37, 8129 (2004); quant-ph/0407195.
[13] R. de la Madrid, Eur. J. Phys. 26, 287 (2005); quant-ph/0502053.
[14] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3949 (2006); quant-ph/0603176.
[15] R. de la Madrid, J. Phys. A: Math. Gen. 39, 3981 (2006); quant-ph/0603177.
[16] J.R. Taylor, Scattering theory, John Wiley & Sons, Inc., New York (1972).
[17] A. Bohm, P. Kielanowski, S. Wickramasekara, “Complex energies and beginnings of time suggest
a theory of scattering and decay,” quant-ph/0510060.
[18] A. Bohm, M. Gadella, Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture Notes
in Physics 348, Berlin (1989).
[19] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. II, Academic Press, New York
(1975).
[20] Y. Strauss, L.P. Hortwitz, A. Volovick, “Approximate resonance states in the semigroup
decomposition of resonance evolution,” quant-ph/0612027.
http://arxiv.org/abs/quant-ph/0109154
http://arxiv.org/abs/quant-ph/0110165
http://arxiv.org/abs/quant-ph/0210167
http://arxiv.org/abs/quant-ph/0407195
http://arxiv.org/abs/quant-ph/0502053
http://arxiv.org/abs/quant-ph/0603176
http://arxiv.org/abs/quant-ph/0603177
http://arxiv.org/abs/quant-ph/0510060
http://arxiv.org/abs/quant-ph/0612027
	Introduction
	The ``standard method''
	TAQT vs. the ``standard method''
	The Quantum Arrow of Time (QAT)
	TAQT vs. the ``classic results''
	Further remarks
	Conclusions

  In this reply, we show that when we apply standard distribution theory to the
Lippmann-Schwinger equation, the resulting spaces of test functions would
comply with the Hardy axiom only if classic results of Paley and Wiener, of
Gelfand and Shilov, and of the theory of ultradistributions were wrong. As
well, we point out several differences between the ``standard method'' of
constructing rigged Hilbert spaces in quantum mechanics and the method used in
Time Asymmetric Quantum Theory.

Introduction
The authors of [1] allege to have shown that the conclusions of [2] regarding the
inconsistency of Time Asymmetric Quantum Theory (TAQT) with quantum mechanics
are false. In this reply, we will show that the arguments of [1] are missing essential
aspects of [2], and that therefore the conclusions of [2] still stand.
The most important claims of [1] are the following:
1. There are many examples of TAQT, and the present author has inadvertently
constructed another one.
2. The flaws of the Quantum Arrow of Time (QAT) pointed out in [2] are actually not
flaws, because the original derivation of the QAT was misquoted from its source [3].
3. The crucial argument of [2] regarding the exponential blowup of the test functions
ϕ̂±(z) does not prevent ϕ̂±(z) from being of Hardy class.
As we shall see, all these claims do not stand close scrutiny. In order to show why,
in Sec. 2 we will outline the method to construct rigged Hilbert spaces in quantum
mechanics based on the theory of distributions [4]. We shall refer to this method as
the “standard method” and show that the resulting rigged Hilbert spaces are not of
Hardy class. We shall also explain the meaning of the exponential blowup of ϕ̂±(z) and
why it implies that the spaces of test functions are not of Hardy class. In Sec. 3, we
briefly outline the method to introduce rigged Hilbert spaces of Hardy class in TAQT
and compare such method with the “standard method.” It will then be apparent that
using the method of TAQT, one can introduce any arbitrary rigged Hilbert space for the
http://arxiv.org/abs/0704.1613v1
Gamow states. In order to address claim 2, we show (again) in Sec. 4 that no matter
how one introduces it, the Quantum Arrow of Time has little to do with the actual time
evolution of a quantum system. To address claim 3, in Sec. 5 we use classic results of
Paley and Wiener and of Gelfand and Shilov to show that the “standard method” of
dealing with the Lippmann-Schwinger equation leads to rigged Hilbert spaces that are
not of Hardy class. Section 7 concludes that the arguments of [2] still stand.
2. The “standard method”
In this section, we illustrate the main features of the “standard method” to construct
rigged Hilbert spaces in quantum mechanics [5]. Such “standard method” is based on
the theory of distributions [4]. For the sake of clarity, we shall use the spherical shell
potential of height V0,
V (~x) = V (r) =
0 0 < r < a
V0 a < r < b
0 b < r < ∞ .
(2.1)
For l = 0, the Hamiltonian acts as (we take ~2/2m = 1)
H = − d
+ V (r) . (2.2)
The regular solution is
χ(r;E) =
E r) 0 < r < a
J1(E)ei
E−V0 r + J2(E)e−i
E−V0 r a < r < b
J3(E)ei
E r + J4(E)e−i
E r b < r < ∞ .
(2.3)
The Jost functions and the S matrix are given by
J+(E) = −2iJ4(E) , J−(E) = 2iJ3(E) , (2.4)
S(E) =
J−(E)
J+(E)
. (2.5)
The solutions of the Lippmann-Schwinger equation can be written as
〈r|E±〉 ≡ χ±(r;E) =
χ(r;E)
J±(E)
. (2.6)
When V tends to zero, these eigensolutions tend to the “free” eigensolution:
〈r|E〉 ≡ χ0(r;E) =
E r) . (2.7)
These eigenfunctions are delta-normalized and therefore their associated unitary
operators,
(U±f)(E) =
dr χ±(r;E) f(r) ≡ f̂±(E) , E ≥ 0 , (2.8)
(U0f)(E) =
dr χ0(r;E) f(r) ≡ f̂0(E) , E ≥ 0 , (2.9)
transform from L2([0,∞), dr) onto L2([0,∞), dE).
The Lippmann-Schwinger and the “free” eigenfunctions can be analytically
continued from the scattering spectrum into the whole complex plane. We shall denote
such analytically continued eigenfunctions by χ±(r; z) and χ0(r; z). Whenever they
exist, the analytic continuations of (2.8) and (2.9) are denoted by
f̂±(z) =
dr χ±(r; z) f(r) , (2.10)
f̂0(z) =
dr χ0(r; z) f(r) , (2.11)
where here and in the following z belongs to a two-sheeted Riemann surface.
The resonant energies are given by the poles zn of the S matrix, and their associated
Gamow states are
u(r; zn) = Nn
J3(zn) sin(
zn r) 0 < r < a
J1(zn)
J3(zn)e
zn−V0 r +
J2(zn)
J3(zn)e
zn−V0 r a < r < b
zn r b < r < ∞ ,
(2.12)
where Nn is a normalization factor.
The theory of distributions [4] says that a test function ϕ(r) on which a distribution
d(r) acts is such that the following integral is finite:‡
〈ϕ|d〉 ≡
dr ϕ(r)d(r) < ∞ , (2.13)
where 〈ϕ|d〉 represents the action of the functional |d〉 on the test function ϕ. With some
variations, this is the “standard method” followed by [7–15] to introduce spaces of test
functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the
method followed by the present author runs (somewhat) parallel to [8], not to TAQT.
In order to use (2.13) to construct the rigged Hilbert spaces for the analytically
continued Lippmann-Schwinger eigenfunctions and for the Gamow states, we need to
obtain the growth of χ±(r; z), χ0(r; z) and u(r; zn). Because the regular solution blows
up exponentially [16],
|χ(r; z)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r , (2.14)
the growth of the eigenfunctions (2.6), (2.7) and (2.12) blows up exponentially:
|χ±(r; z)| ≤ C 1
J±(z)
|z|1/4 r
1 + |z|1/2 r
z |r , (2.15)
‡ In quantum mechanics, we need to impose a few more requirements, but we will not need to go into
such details here.
|χ0(r; z)| ≤ C
|z|1/4 r
1 + |z|1/2 r
z |r , (2.16)
|u(r; zn)| ≤ Cn
|zn|1/2 r
1 + |zn|1/2 r
zn |r . (2.17)
When we plug this exponential blowup into the basic requirement (2.13) of the “standard
method,” we see that the test functions on which those distributions act must fall off at
least exponentially.
By using the Gelfand-Shilov theory of M an Ω functions [4], it was shown in [15]
that when a and b are positive real numbers satisfying
= 1 , (2.18)
and when ϕ+(r) is an infinitely differentiable function whose tails fall off like e−r
a/a, then
ϕ+(z) grows like e|Im(
z)|b/b in the infinite arc of the lower half-plane of the Riemann
surface:
If |ϕ+(r)| < Ce−
a as r → ∞, then |ϕ̂+(z)| ≤ Ce
b as |z| → ∞ . (2.19)
It was shown in [2] that when ϕ+(r) ∈ C∞0 , ϕ̂+(z) blows up exponentially in the infinite
arc of the lower half-plane of the Riemann surface:
If |ϕ+(r)| = 0 when r > A, then |ϕ̂+(z)| ≤ CeA|Im
z | as |z| → ∞ . (2.20)
From the above estimates, we concluded in [2] that the ϕ+’s obtained from the “standard
method” cannot be Hardy functions, since ϕ̂+(z) does not tend to zero as |z| tends to
infinity.
The authors of [1] argue that one cannot draw any conclusion on the limit |z| → ∞
from estimates such as (2.19) or (2.20), and therefore they conclude that nothing
prevents ϕ̂+(z) from tending to zero and therefore from being Hardy functions. Their
conclusion is not true, because their argument does not take the nature of (2.19) and
(2.20) into account. After we explain the meaning of those estimates, it will be clear
why they prevent ϕ̂±(z) from tending to zero in any infinite arc of the Riemann surface.
In order to understand what (2.19) and (2.20) mean, we start with the simple sine
function sin(
E r). When E ≥ 0, the sine function oscillates between +1 and −1:
| sin(
E r)| ≤ 1 , E ≥ 0 . (2.21)
As E tends to infinity, such oscillatory behavior remains, and in such limit the sine
function does not tend to zero. When we analytically continue the sine function,
z r) , (2.22)
the oscillations are bounded by
| sin(
z r)| ≤ C
|z|1/2 r
1 + |z|1/2 r
z |r . (2.23)
Thus, as |z| tends to infinity, sin(
z r) oscillates wildly, and the magnitude of its
oscillation is tightly bounded by the exponential function. It is certain that as |z|
tends to infinity, sin(
z r) does not tend to zero, even though the function vanishes
z r = ±nπ, n = 0, 1, . . .
It just happens that the solutions of the Lippmann-Schwinger equation follow the
same pattern. When E is positive, the eigensolutions are oscillatory and bounded by
|χ±(r;E)| ≤ C 1
J±(E)
|E|1/4 r
1 + |E|1/2 r
. (2.24)
When the energy is complex, their oscillations get wild and are bounded by Eq. (2.15).§
Thus, the analytic continuations of the Lippmann-Schwinger eigenfunctions oscillate
wildly, and the magnitude of their oscillation is tightly bounded by an exponential
function (multiplied by factors that don’t cancel the exponential blowup when |z| → ∞).
Because in Eqs. (2.10) and (2.11) we are integrating over r, the exponentially-
bounded oscillations of χ±(r; z) get transmitted into ϕ̂±(z). The estimates (2.19) and
(2.20) bound the oscillation of the test functions of the “standard method,” except
for factors that don’t cancel the exponential blowup. It is the exponentially-bounded
oscillations of ϕ̂±(z) what prevent ϕ̂±(z) from tending to zero in any infinite arc of the
Riemann surface and therefore from being of Hardy class.
A somewhat simpler way to understand the above estimates is by looking at the
“free” incoming and outgoing wave functions ϕin and ϕout. Because in the energy
representation such wave functions are the same as the “in” and “out” wave functions,
ϕ̂in(E) = 〈E|ϕin〉 = 〈+E|ϕ+〉 = ϕ̂+(E) , (2.25)
ϕ̂out(E) = 〈E|ϕout〉 = 〈−E|ϕ−〉 = ϕ̂−(E) , (2.26)
in TAQT the analytic continuation of ϕ̂in(E) and ϕ̂out(E) are also of Hardy class. Since
ϕ̂in,out(z) =
z r)ϕin,out(r) , (2.27)
it is evident that the exponential blowup (2.23) of sin(
z r) will prevent ϕ̂in,out(z) from
tending to zero as |z| → ∞ in any half-plane of the Riemann surface. Thus, ϕ̂in,out(z)
are not of Hardy class, contrary to TAQT.
Strictly speaking, the bounds (2.19) and (2.20) are not the tightest ones. We should
include polynomial corrections, see Eq. (B.15) in [15], and the effect of
|z|1/4r
1+|z|1/2r
and 1J±(z)
to obtain the tightest bounds. We shall not obtain those corrections here, because they
do not cancel the exponential blowup at infinity, and because in this reply we shall use
instead other classic bounds, see Sec. 5.
Let us summarize this section. In standard quantum mechanics, once the
Lippmann-Schwinger equation is solved, the properties of ϕ̂±(z) are already determined
by Eqs. (2.10) and (2.11), and there is no room for any extra assumption on their
properties. This means, in particular, that the Hardy axiom cannot be simply assumed.
Rather, the Hardy axiom must be proved using Eqs. (2.10) and (2.11).‖ It simply
§ The points at which J±(z) = 0 do not affect the essence of the argument.
‖ This is what in [2] it was meant by the assertion that the Hardy axiom is not a matter of assumption
but a matter of proof.
happens that the “standard method” yields ϕ̂±(z) and ϕ̂in,out(z) that oscillate wildly.
Because these oscillations are bounded by exponential functions, ϕ̂±(z) and ϕ̂in,out(z) do
not tend to zero as |z| tends to infinity in any half-plane of the Riemann surface—hence
they are not of Hardy class.
3. TAQT vs. the “standard method”
In TAQT, one doesn’t solve the Lippmann-Schwinger equation in order to afterward
obtain the properties of ϕ̂±(z) using Eq. (2.10). Instead, one transforms into the energy
representation (using U± in our example) and then imposes the Hardy axiom. If H2±
denotes the spaces of Hardy functions from above (+) and below (−), S denotes the
Schwartz space, and Φ̃± denote their intersection restricted to the positive real line,
Φ̃± = H2± ∩ S|R+ , (3.1)
then the Hardy axiom states that the functions ϕ̂±(z) belong to Φ̃∓:
ϕ̂±(z) ∈ Φ̃∓ . (3.2)
This means that in the position representation, the Gamow states and the analytic
continuation of the Lippmann-Schwinger eigenfunctions act on the following spaces:
ΦBG∓ = U
± Φ̃∓ . (3.3)
It is obvious that the choices (3.2)-(3.3) are arbitrary. One may as well choose another
dense subset of L2([0,∞), dE) with different properties and obtain a different space of
test functions for the Gamow states. What is more, ΦBG± are different from the spaces
of test functions obtained through the “standard method,” because the functions ϕ̂±(z)
of the “standard method” are not of Hardy class.
The authors of [1] claim that the present author has inadvertently constructed
an example of TAQT. That such is not the case can be seen not only from the
differences between the “standard method” and the method used in TAQT to introduce
rigged Hilbert spaces, but also from the outcomes. For example, whereas in the
position representation the “standard method” calls for just one rigged Hilbert
space for the Gamow states and for the analytically continued Lippmann-Schwinger
eigenfunctions [15], TAQT uses two rigged Hilbert spaces
ΦBG± ⊂ L2([0,∞), dr) ⊂ Φ×BG± . (3.4)
One of the rigged Hilbert spaces is used for the “in” solutions and for the anti-resonant
states, whereas the other one is used for the “out” solutions and for the resonant states.
Another difference is that in TAQT, the solutions of the Lippmann-Schwinger equation
for scattering energies have a time asymmetric evolution [17], whereas the “standard
method” yields that such time evolution runs from t = −∞ to t = +∞, see [14].
Incidentally, this is an instance where TAQT differs not only mathematically but also
physically from standard quantum mechanics, because in standard scattering theory,
the time evolution of a scattering process goes from the asymptotically remote past
(t → −∞) to the asymptotically far future (t → +∞). This is not so in TAQT [17].
It seems hardly necessary to clarify what the present author means by “standard
quantum mechanics.” Standard quantum mechanics means the Schrödinger equation,
and standard scattering theory means the Lippmann-Schwinger equation. In standard
quantum mechanics, one assumes that these equations describe the physics and then
solves them. Because of the scattering and resonant spectra, their solutions lie within
rigged Hilbert spaces. The construction of such rigged Hilbert spaces follows by
application of the “standard method.” By contrast, TAQT simply assumes that the
solutions of the Schrödinger and the Lippmann-Schwinger equations comply with the
Hardy axiom, without ever showing that the actual solutions of those equations comply
with such axiom.
It was claimed in [2] that there is no example of TAQT. The authors of [1] dispute
such claim and assert that there are many examples. The present author disagrees
with their assertion, because assuming that for a large class of potentials the solutions
of the Lippmann-Schwinger equation comply with the Hardy axiom is not the same
as having an example where it is shown that the actual solutions of the Lippmann-
Schwinger equation comply with the Hardy axiom. In fact, to the best of the present
author’s knowledge, no advocate of TAQT has ever used Eq. (2.10) to discuss the
analytic properties of ϕ̂±(E) = 〈±E|ϕ±〉 in terms of the actual solutions χ±(r;E) of
the Lippmann-Schwinger equation.
The authors of [1] inadvertently acknowledge that there is no example of TAQT
when they say that they still need “to identify the form and properties” of the functions
of (3.3), see the last paragraph in section 2 of [1]. By saying so, they are acknowledging
that they don’t know whether the standard Gamow states defined in the position
representation are well defined as functionals acting on ΦBG±. If TAQT had an example,
it would be known.
4. The Quantum Arrow of Time (QAT)
Advocates of TAQT argue that their choice (3.3) is not arbitrary but rather is rooted
on a causality principle. Such causality principle is the “preparation-registration arrow
of time,” sometimes referred to as the “Quantum Arrow of Time” (QAT). For the “in”
states ϕ+, the causal statement of the QAT is written as
ϕ̃+(t) ≡
dE e−iEtϕ̂+(E) = 0 , for t > 0 . (4.1)
By one of the Paley-Wiener theorems, Eq. (4.1) is equivalent to assuming that ϕ̂+(E)
is of Hardy class from below. The corresponding causal statement for the “out” wave
functions ϕ− implies that ϕ− is of Hardy class from above. Hence, in TAQT, the
choice (3.3) is not arbitrary but a consequence of causality.
It was pointed out in [2] that the QAT is flawed. The argument was twofold. First,
it was pointed out that the original derivation [3] of Eq. (4.1) made use of the following
flawed assumption:
0 = 〈E|ϕin(t)〉 = 〈+E|ϕ+(t)〉 = e−iEtϕ̂+(E) , for all energies, (4.2)
which can happen only when ϕ+ and ϕin are identically 0. It was then pointed out that
even though one may simply assume the causal statement (4.1) and forget about how
it was derived, such causal statement says little about the actual time evolution of a
quantum system, because the quantum mechanical time evolution of ϕ+ is not given by
Eq. (4.1):
ϕ+(t) = e−iHtϕ+ 6= ϕ̃+(t) . (4.3)
To counter this argument, the authors of [1] claim that the derivation of the QAT
was misquoted from the original source [3], and that the flawed assumption (4.2) was
never used to derive the QAT (4.1). It seems therefore necessary to quote the original
derivation (see [3], page 2597):¶
“We are now in the position to give a mathematical formulation of the QAT: we
choose t = 0 to be the time before which all preparations of φin(t) are completed and
after which the registration of ψout(t) begins. This means that for t > 0 the energy
distribution of the preparation apparatus must vanish: 〈E, η|φin(t)〉 = 0 for all values
of the quantum numbers E and η (η are the additional quantum numbers which we
usually suppress). As the mathematical statement for ‘no preparations for t > 0’ we
therefore write (the slightly weaker condition)
dE 〈E|φin(t)〉 =
dE 〈+E|φ+(t)〉 =
dE 〈+E|e−iHt|φ+〉 (4.4)
dE 〈+E|φ+〉e−iEt ≡ F(t) for t > 0 . (4.5)
The readers can decide whether or not the flawed hypothesis (4.2) was used to derive
the QAT (4.5).
Nevertheless, it is actually not very relevant whether the authors of [3] used (4.2)
to derive (4.1). As pointed out in [2], and as mentioned above, even though one can
forget (4.2) and simply assume (4.1) as the causal condition to be satisfied by ϕ+, such
causal condition has little to do with the time evolution of a quantum system, see again
Eq. (4.3). In particular, as even the author of [6] has asserted, the t that appears in
Eq. (4.1) is not the same as the parametric time t that labels the evolution of a quantum
system.+ Thus, as far as standard quantum mechanics is concerned, the causal content
of the QAT is physically vacuous, and therefore, regardless of how one motivates it,
there is no physical justification for the choice (3.3).
5. TAQT vs. the “classic results”
In this section, we are going to compare the Hardy axiom of TAQT with some
classic results of Paley and Wiener, of Gelfand and Shilov and of the theory of
ultradistributions, which we shall collectively refer to as the “classic results.” More
¶ In this quote, φin, φ+, F(t) and Eq. (4.5) correspond, respectively, to ϕin, ϕ+, ϕ̃+(t) and Eq. (4.1).
+ All this shows that the new term TAQT is a misnomer. A better name is Bohm-Gadella theory,
because it was these two authors who proposed the theory and summarized it in [18].
precisely, we will see that the spaces of test functions ϕ̂± obtained by the “standard
method” would be of Hardy class only if the “classic results” were wrong.
The direct comparison with the “classic results” is more easily done in one
dimension, and therefore we shall use the example of the one-dimensional rectangular
barrier potential:
V (x) =
0 −∞ < x < a
V0 a < x < b
0 b < x < ∞ .
(5.1)
For this potential, the “in” and “out” eigensolutions are well known and can be found
for example in [12]. We shall denote them by χ±l,r(x;E), where the labels l,r denote
left and right incidence. When we analytically continue these eigenfunctions, or when
we consider the Gamow states for this potential, the “standard method” calls for test
functions ϕ±l,r(x) for which the following integrals are finite:
ϕ̂±l,r(z) =
dxχ±l,r(x; z)ϕ(x) . (5.2)
Just as in the example discussed in Sec. 2, the test functions ϕ(x) must at least fall off
faster than exponentials.
To further simplify the discussion, we need to recall that, because of Eqs. (2.25)
and (2.26), the Hardy axiom assumes that the “free” wave functions ϕ̂inl,r(E) and ϕ̂
l,r (E)
are also of Hardy class. These “free” functions are given by (hereafter, we just consider
ϕinl,r, since the analysis for ϕ
l,r is the same)
ϕ̂inl (E) =
dx e−ikx ϕin(x) , (5.3)
ϕ̂inr (E) =
dx eikx ϕin(x) , (5.4)
where k =
E is the wave number. The total wave function is given by the sum of left
and right components:
ϕ̂in(E) = ϕ̂inl (E) + ϕ̂
r (E) . (5.5)
It is simpler to work with k rather than with E and define
ϕ̂inl,r(k) ≡
2k ϕ̂inl,r(E) ; (5.6)
that is,
ϕ̂inl (k) =
dx e−ikx ϕin(x) , k ≥ 0 , (5.7)
ϕ̂inr (k) =
dx eikx ϕin(x) , k ≥ 0 . (5.8)
The “total” wave function in the wave-number representation, ϕ̂in(k) = ϕ̂inl (k)+ ϕ̂
r (k),
is thus the Fourier transform of ϕ(x),
ϕ̂in(k) =
dx e−ikx ϕin(x) , k ∈ R . (5.9)
Its analytic continuation will be denoted as
ϕ̂in(q) =
dx e−iqx ϕin(x) , q ∈ C . (5.10)
At this point, we are ready to introduce two classic theorems. The first one is due
to Paley and Wiener (see Theorem IX.11 in [19]):
Theorem 1 (Paley-Wiener). An entire analytic function ϕ̂(q) is the Fourier
transform of a C∞0 (R) function ϕ(x) with support in the segment {x | |x| < A} if,
and only if, for each N there is a CN so that
|ϕ̂(q)| ≤ CN e
A|Im(q)|
(1 + |q|)N
(5.11)
for all q ∈ C.
This theorem says that the Fourier transform of a C∞0 function is an analytic
function that grows exponentially, and that such exponential growth is mildly corrected
(but not canceled) by a polynomial falloff.
The second theorem we shall use is due to Gelfand and Shilov [4]. Before stating it,
we need some definitions. Let a and b denote two positive real numbers satisfying (2.18).
Let us define Φa,b as the set of all differentiable functions ϕ(x) (−∞ < x < ∞) satisfying
the inequalities
dnϕ(x)
∣∣∣∣ ≤ Cne
−α |x|
a (5.12)
with constants Cn and α > 0 which may depend on the function ϕ. Let us define the
space Φ̂a,b as the set of entire analytic functions ϕ̂(q), q = Re(q) + i Im(q), which satisfy
the inequalities
|qnϕ̂(q)| ≤ Cne+β
|Im(q)|b
b , (5.13)
where the constants Cn and β > 0 depend on the function ϕ. It is obvious that
the elements of Φa,b are functions that, together with their derivatives, decrease at
infinity faster than e−
a , whereas the elements of Φ̂a,b are analytic functions that grow
exponentially at infinity as e+
|Im(q)|b
b , except for a polynomial correction that doesn’t
cancel the exponential blowup.
Theorem 2 (Gelfand-Shilov). The space Φ̂a,b is the Fourier transform of Φa,b.
This theorem means that the smooth functions that fall off at infinity faster than
e−|x|
a/a are, in Fourier space, analytic functions that grow exponentially like e+|Im(q)|
The bounds (5.11) and (5.13) are to be understood in the same way as the
bounds (2.19) and (2.20). That is, the bounds (5.11) and (5.13) mean that ϕ̂(q) is
an oscillatory function that grows exponentially in the infinite arc of the q-plane, the
oscillation being tightly bounded by Eqs. (5.11) and (5.13) when ϕ(x) belongs to C∞0
and Φa,b, respectively. Note that after the addition of the corresponding polynomial
corrections, the bounds (2.19) and (2.20) are entirely analogous to the bounds (5.11)
and (5.13)—the operators U± are after all Fourier-like transforms [12].
Let us now apply the above theorems to the functions ϕin(x) obtained by the
“standard method.” In order for Eq. (5.10) to make sense, ϕin(x) must fall off faster
than exponentials. If we choose ϕin(x) to fall off like e−|x|
a/a, then the Gelfand-Shilov
theorem tells us that ϕ̂in(q) grows like e+|Im(q)|
b/b. Even when we impose that ϕin(x)
is C∞0 , which is already a very strict requirement, the Paley-Wiener theorem says that
ϕ̂in(q) grows exponentially. This means, in particular, that the ϕ̂in(q) do in general not
tend to zero in the infinite arc of the q-plane, because if they did, the Paley-Wiener
and the Gelfand-Shilov theorems would be wrong. Because of Eq. (5.6), ϕ̂in(z) does in
general not tend to zero as |z| tends to infinity in the lower half-plane of the second
sheet. Hence the space of ϕ̂in’s is not of Hardy class from below.
The space of ϕ̂+’s cannot be of Hardy class from below either, because if it were,
|z|→∞
ϕ̂+(z) = 0 , (5.14)
where the limit is taken in the lower half plane of the second sheet. By Eq. (2.25),
this implies that also the space of ϕ̂in’s would be of Hardy class and comply with this
limit, which we know is not possible due to the “classic results.” Thus, the “standard
method” yields spaces of test functions that do not comply with the Hardy axiom. This
is precisely what it was meant in [2] by the assertion that TAQT is inconsistent with
standard quantum mechanics.
To finish this section, we note that if we chose the test functions as in [8], then
we would be dealing with ultradistributions. In Fourier space, the test functions for
ultradistributions grow faster than any exponential as we follow the imaginary axis,
see [8] and references therein. Thus, if the “standard method” yielded spaces of Hardy
functions, that property of ultradistributions would be false.
6. Further remarks
The authors of [1] claim that it is inaccurate to state that the proponents of TAQT
dispense with asymptotic completeness. This statement should be compared with the
first quote in section 6 of [2].
The authors of [1] also claim that TAQT obtains the resonant states by solving
the Schrödinger equation subject to purely outgoing boundary conditions. This claim
should be compared with the second quote in section 6 of [2].
The authors of [1] also dispute the assertion of [2] that TAQT sometimes uses the
whole real line as though it coincided with the scattering spectrum of the Hamiltonian.
A glance at, for example, the QAT (4.1) seems to support such assertion.
7. Conclusions
In standard scattering theory, one assumes that the physics is described by the
Lippmann-Schwinger equation. When one solves such equation, one finds that its
solutions must be accommodated by a rigged Hilbert space, and that its time evolution
runs from t = −∞ till t = +∞ [14]. When one analytically continues the solutions
of the Lippmann-Schwinger equation, one finds that they must be accommodated by
one rigged Hilbert space, which also accommodates the resonant (Gamow) states. The
construction of such rigged Hilbert space is determined by standard distribution theory.
By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger
equations belong to two rigged Hilbert spaces of Hardy class. In TAQT, one never
explicitly solves the Lippmann-Schwinger equation for specific potentials in the position
representation. Instead, one assumes that its solutions satisfy the Hardy axiom. Unlike
in standard scattering theory, in TAQT the time evolution of the solutions of the
Lippmann-Schwinger equation does not run from t = −∞ till t = +∞.
By comparing the properties of the actual solutions of the Lippmann-Schwinger
equation with the Hardy axiom, we have seen that such actual solutions would comply
with the Hardy axiom only if classic results of Paley and Wiener, of Gelfand and Shilov,
and of the theory of ultradistributions were wrong. We have (again) stressed the fact
that the Quantum Arrow of Time, which is the justification for using the rigged Hilbert
spaces of Hardy class, has little to do with the time evolution of a quantum system. We
have stressed that using the method of TAQT to introduce rigged Hilbert spaces, we
could accommodate the Gamow states in a landscape of arbitrary rigged Hilbert spaces,
see also [5].
Our claim of inconsistency should not be taken as a claim that TAQT is
mathematically inconsistent or that TAQT doesn’t have a beautiful mathematical
structure. What the present author claims is that TAQT is not applicable in quantum
mechanics and is in fact a different theory.
To finish, we would like to mention that the “classic theorems” are not in conflict
with using Hardy functions in quantum mechanics. They are in conflict only with the
Hardy axiom. Thus, our results do not apply to other works that use Hardy functions
in a different way [20].
Acknowledgment
This research was supported by MEC and DOE.
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http://arxiv.org/abs/quant-ph/0109154
http://arxiv.org/abs/quant-ph/0110165
http://arxiv.org/abs/quant-ph/0210167
http://arxiv.org/abs/quant-ph/0407195
http://arxiv.org/abs/quant-ph/0502053
http://arxiv.org/abs/quant-ph/0603176
http://arxiv.org/abs/quant-ph/0603177
http://arxiv.org/abs/quant-ph/0510060
http://arxiv.org/abs/quant-ph/0612027
	Introduction
	The ``standard method''
	TAQT vs. the ``standard method''
	The Quantum Arrow of Time (QAT)
	TAQT vs. the ``classic results''
	Further remarks
	Conclusions
