Theory Including Future Not Excluded -- Formulation of Complex Action Theory II

0.56 MB·English

by  Keiichi Nagao

IU-TH-9
Theory Including Future Not Excluded
Formulation of Complex Action Theory II
3
1
0 Keiichi Nagao ) and Holger Bech Nielsen )
2 ∗ ∗∗

n
a )Faculty of Education, Ibaraki University, Mito 310-8512 Japan
∗
J

7 and
1
] ∗∗) Niels Bohr Institute, University of Copenhagen, 17 Blegdamsvej
h
p Copenhagen φ, Denmark
-
t
n
a
u
q
[

3
Abstract
v
6
0 We study a complex action theory (CAT) whose path runs over not only past but
7
3 also future. We show that if we regard a matrix element defined in terms of the future
.
5 state at time T and the past state at time T as an expectation value in the CAT,
B A
0
2 then we are allowed to have the Heisenberg equation, the Ehrenfest’s theorem and
1
the conserved probability current density. In addition we show that the expectation
:
v
i value at the present time t of a future-included theory for large T t and large
X B −
t T corresponds to that of a future-not-included theory with a proper inner product
r A
a −
for large t T . Hence the CAT with future explicitly present in the formalism and
A
−
influencing in principle the past is not excluded phenomenologically, because the effects
are argued to be very small in the present era. Furthermore we explicitly derive the
Hamiltonian for the future state via path integral, and confirm that it is given by the
hermitian conjugate of the Hamiltonian for the past state.
∗) E-mail: [email protected]
∗∗) E-mail: [email protected]
1 typeset using TEX.cls Ver.0.89
PTP h i
§1. Introduction
Quantum theories are properly formulated via the Feynman path integral (FPI). Usually
the action S is real, and it is thought to be more fundamental than the integrand exp(iS).
~
However, if we assume that the integrand is more fundamental than the action in quantum
theory, then it is naturally thought that since the integrand is complex, the action could also
be complex. Based on this speculation and other related works in some backward causation
developments inspired by general relativity1) and the non-locality explanation of fine-tuning
problems,2) the complex action theory (CAT) has so far been studied intensively.3),4) The
imaginary part of the action is thought to give some falsifiable predictions, and many inter-
estingsuggestionshavebeenmadefortheHiggsmass,5) quantum-mechanical philosophy,6)–8)
some fine-tuning problems,9),10) black holes,11) the de Broglie-Bohm particle, and a cut-off
in loop diagrams.12)
In refs.3)–12) they studied a future-included version, i.e., the theory including not only
a past time but also a future time as an integration interval of time. In contrast to these
works, in refs.13)–15) we studied a future-not-included version. In ref.13) we analyzed the
time development of some state by a non-Hermitian diagonalizable bounded Hamiltonian
H, and showed that we can effectively obtain a Hermitian Hamiltonian after a long time
development by introducing a proper inner product ) based on the speculation in ref.17) If
∗
the Hermitian Hamiltonian is given in a local form, a conserved probability current density
can be constructed with two kinds of wave functions. We note that the non-Hermitian
Hamiltonianstudied there isageneric one, so it doesnotbelong to theclass ofPTsymmetric
non-Hermitian Hamiltonians, which has recently been intensively studied.16),18)
In ref.14) introducing a philosophy to keep the analyticity in the parameter variables of
FPI and defining a modified set of a complex conjugate, real and imaginary parts, Hermitian
conjugates, and bras, we explicitly constructed non-Hermitian operators of coordinate and
momentum, qˆ and pˆ , and their eigenstates q and p for complex q and
new new m new m new
h | h |
p by utilizing coherent states of harmonic oscillators so that we can deal with complex q
and p. In addition, applying this complex coordinate formalism to the study of ref.,13) we
showed that the mechanism for suppressing the anti-Hermitian part of the Hamiltonian after
a long time development also works in the complex coordinate case. In ref.15) based on the
complex coordinate formalism we explicitly examined the definitions of the momentum and
Hamiltonian via FPI, and confirmed that they have the same forms as those in the real
action theory (RAT).
Regarding other studies related to complex coordinates, in refs.19),20) the complete set
∗) A similar inner product was also studied in ref.16)
2
of solutions of the differential equations following from the Schwinger action principle has
been obtained by generalizing the path integral to include sums over various inequivalent
contours of integration in the complex plane. In ref.,21) complex Langevin equations have
been studied, and in refs.22)23) a method to examine the complexified solution set has been
investigated.
Inthispaperwestudyafuture-includedversionoftheCATwhosepathrunsovernotonly
past but also future3) using both the complex coordinate formalism14) and the mechanism
for suppressing the anti-Hermitian part of the Hamiltonian.13) In ref.,3) one of the authors
of this paper, H.B.N., and Ninomiya introduced not only the ordinary past state A(T ) at
A
| i
the initial time T , but also a future state B(T ) at the final time T , where T and T
A B B A B
| i
are set to be and respectively. Here A(T ) and B(T ) time-develop according to
A B
−∞ ∞ | i | i
the non-Hermitian Hamiltonian Hˆ and Hˆ , respectively, where Hˆ is set to be equal to Hˆ .
B B †
They studied the matrix element of some operator defined by
O
B(t) A(t)
BA h |O| i, (1.1)
hOi ≡ B(t) A(t)
h | i
where t is the present time. In the RAT, such a future state as B is already introduced
| i
in ref.24) in a different context. The matrix element of eq.(1.1), which is called the weak
value, has also been intensively studied. For details of the weak value, see the reviews25)
and references therein. Eq.(1.1) is a matrix element in the usual sense, but in a future-
included version of the CAT we speculate that it can be regarded as the expectation value
of from the results that we obtain in this paper. As we shall see later, BA allows us
O hOi
to have the Heisenberg equation. In addition, we shall confirm that it gives us Ehrenfest’s
theorem. Furthermore, we shall also see that a conserved probability current density can be
constructed. Therefore we regard it as an expectation value in the future-included theory.
Here we note that since the future-included theory differs from ordinary quantum me-
chanics on two points – the existence of the imaginary part of the action S and that of
the future state – it seems excluded phenomenologically. So it is necessary that the future-
included theory is not excluded, to show that usual physics is approximately obtained from
it. Indeed, in ref.,3) an attempt was made to obtain a correspondence between the future-
included theory and ordinary quantum mechanics, and it is speculated that BA becomes
hOi
A(t) A(t)
AA h |O| i, (1.2)
hOi ≡ A(t) A(t)
h | i
i.e. the expectation value of in the future-not-included theory. We review this speculation
O
andmakeitclearthattherearepointstobeimprovedintheargument. Thenwestudy BA
hOi
with more care concerning the inner product being obtained by using both the complex
3
coordinate formalism and the mechanism for suppressing the anti-Hermitian part of the
Hamiltonian, and show that BA becomes an expectation value with a different inner
hOi
product defined in the future-not-included theory. Next we show that the inner product can
beinterpreted asoneofthepossible proper inner productsrealizedinthefuture-not-included
theory. Thus we shall have the correspondence principle: thefuture-included theory for large
T t and large t T is almost equivalent to the future-not-included theory for large t T ,
B A A
− − −
which means that such theories with complex action and functional integral of future time
are not excluded. Incidentally, as for the Hamiltonians in the future-included theory, there
are two Hamiltonians Hˆ and Hˆ , but only Hˆ is derived in ref.15) Therefore, in this paper
B
we give the explicit derivation of Hˆ via the path integral using the method in ref.,15) and
B
confirm that it is given by Hˆ = Hˆ .
B †
This paper is organized as follows. In section 2 we review our complex coordinate formal-
ism and give a theorem for matrix elements. In section 3 we review the proper inner product
for the Hamiltonian Hˆ, and introduce another proper inner product for the Hamiltonian
Hˆ . Next we review the mechanism for suppressing the anti-Hermitian part of Hˆ after a
B
long time development. In section 4 we study the various properties of the expectation value
BA. We show that it allows us to have the Heisenberg equation, Ehrenfest’s theorem
hOi
and a conserved probability current density. In section 5 after reviewing the study in ref.,3)
we show that the expectation value of the future-included theory for large T t and large
B
−
t T corresponds to that of the future-not-included theory for large t T with a proper
A A
− −
inner product. Section 6 is devoted to the summary and outlook. In appendix A we give an
explicit derivation of Hˆ via the path integral following ref.15)
B
§2. Review of the complex coordinate formalism
In this section we briefly review the complex coordinate formalism that we proposed in
ref.14) so that we can deal with the complex coordinate q and momentum p properly in the
CAT. We emphasize that even in a real action theory (RAT) we encounter complex q and p
at the saddle point in the cases of tunneling effect or WKB approximation, etc. As a simple
and clear example, let us consider a wave function,
ψ(q) = q ψ . (2.1)
h | i
This is defined for real q, but what happens for complex q in the cases mentioned above?
There are no problems with the left-hand side, because we can just say that the function
ψ is analytically extended to complex q. But the right-hand side cannot be extended to
complex q, because q is defined only for real q. Indeed q obeys q qˆ = q q, so if we
h | h | h | h |
4
attempt to extend the real eigenvalue q to complex, we encounter a contradiction because
qˆ is a Hermitian operator. Therefore qˆ and q have to be appropriately extended to a
h |
non-Hermitian operator and its eigenstate for complex q.
2.1. Non-Hermitian operators qˆ and pˆ , and the eigenstates of their Hermitian conju-
new new
gates q and p
new new
| i | i
Following ref.,14) we summarize the construction of the non-Hermitian operators of co-
ordinate and momentum, qˆ and pˆ , and the eigenstates of their Hermitian conjugates
new new
q and p such that
new new
| i | i
qˆ q = q q , (2.2)
n†ew| inew | inew
pˆ p = p p , (2.3)
†new| inew | inew
[qˆ ,pˆ ] = i~, (2.4)
new new
for complex q and p by formally utilizing two coherent states. Our proposal is to replace
qˆ, pˆ, q and p with qˆ , pˆ , q and p . The explicit expressions for qˆ , pˆ ,
| i | i n†ew †new | inew | inew new new
q and p are given by
new new
| i | i
1 pˆ
qˆ qˆ i , (2.5)
new
≡ 1 m′ω′ (cid:18) − mω(cid:19)
− mω
q 1 mω
pˆ pˆ ′ ′qˆ , (2.6)
new
≡ 1 m′ω′ (cid:18) − i (cid:19)
− mω
q 1
|qinew ≡ 4mπω~ 1− mm′ωω′ 4 e−m4~ω(cid:16)1−mm′ωω′(cid:17)q2|sm2~ω 1− mm′ωω′ qicoh, (2.7)
(cid:26) (cid:18) (cid:19)(cid:27) (cid:18) (cid:19)
1
|pinew ≡ 41π−~mmm′′ωωω′′!4 e−4~m1′ω′(cid:16)1−mm′ωω′(cid:17)p2|is2~m1′ω′ (cid:18)1− mm′ωω′(cid:19)picoh′, (2.8)
where λ is a coherent state parametrized with a complex parameter λ defined up to
coh
| i
a normalization factor by |λicoh ≡ eλa†|0i = ∞n=0 √λnn!|ni, and this satisfies the relation
a λ = λ λ . Here a = mω qˆ+i pˆ and a = mω qˆ i pˆ are annihilation
| icoh | icoh 2~ mω P † 2~ − mω
and creation operators, where qˆ and pˆ are the usual Hermitian operators of coordinate and
p (cid:0) (cid:1) p (cid:0) (cid:1)
momentum obeying
qˆq = q q , (2.9)
| i | i
pˆp = p p , (2.10)
| i | i
[qˆ,pˆ] = i~ (2.11)
5
for real q and p. In eq.(2.8) λ is another coherent state, which is defined similarly with
coh′
| i
different parameters m′ω′, |λicoh′ ≡ eλa′†|0i, where a′† is given by a′† = m2′~ω′ qˆ−imp′ˆω′ .
Before seeing the properties of qˆ , pˆ , q , and p , we define aqdelta function of
new new new new (cid:0) (cid:1)
| i | i
complex parameters in the next subsection.
2.2. The delta function
For our later convenience we first define as a class of distributions depending on one
D
complex variable q C. Using a function g : C C as a distribution in the class , we
∈ → D
define the following functional G
G[f] = f(q)g(q)dq (2.12)
ZC
for any analytical function f : C C with convergence requirements such that f 0 for
→ →
q . The functional G is a linear mapping from the function f to a complex number.
→ ±∞
Since the simulated function g is supposed to be analytical in q, the path C, which is chosen
to run from to in the complex plane, can be deformed freely and so it is not relevant.
−∞ ∞
As an approximation to such a distribution we could use the smeared delta function defined
for complex q by
1 2
g(q) = δcǫ(q) ≡ 4πǫe−q4ǫ, (2.13)
r
where ǫ is a finite small positive real number. For the limit of ǫ 0 g(q) converges in the
→
distribution sense for complex q obeying the condition
L(q) (Re(q))2 (Im(q))2 > 0. (2.14)
≡ −
For any analytical test function f(q) ) and any complex q this δǫ(q) satisfies
∗ 0 c
f(q)δǫ(q q )dq = f(q ), (2.15)
c − 0 0
ZC
as long as we choose the path C such that it runs from to in the complex plane and
−∞ ∞
at any q its tangent line and a horizontal line form an angle θ whose absolute value is within
π to satisfy the inequality (2.14). An example of permitted paths is shown in fig.1, and the
4
domain of the delta function is drawn in fig.2.
Next we extend the delta function to complex ǫ, and consider
δcǫ(aq) = 41πǫe−41ǫa2q2 (2.16)
r
∗) BecauseoftheLiouvilletheoremiff isaboundedentirefunction,f isconstant. Soweareconsidering
as f an unbounded entire function or a function that is not entire but is holomorphic at least in the region
on which the path runs.
6
Fig. 1. An example of permitted paths
for non-zero complex a. We express ǫ, q, and a as ǫ = r eiθǫ, q = reiθ, and a = r eiθa. The
ǫ a
convergence condition of δǫ(aq): Re a2q2 > 0 is expressed as
c ǫ
(cid:16) (cid:17)
π 1 π 1
+ (θ 2θ ) < θ < + (θ 2θ ), (2.17)
ǫ a ǫ a
−4 2 − 4 2 −
3 1 5 1
π + (θ 2θ ) < θ < π + (θ 2θ ). (2.18)
ǫ a ǫ a
4 2 − 4 2 −
For q, ǫ, and a such that eqs.(2.17)(2.18) are satisfied, δǫ(aq) behaves well as a delta function
c
of aq, and we obtain the relation
sign(Rea) ǫ
δǫ(aq) = δa2(q), (2.19)
c a c
where we have introduced an expression
1 for Re(a) > 0,
sign(Rea) (2.20)
≡ ( 1 for Re(a) < 0.
−
2.3. New devices to handle complex parameters
In this subsection, to keep the analyticity in dynamical variables of FPI such as q and p
we define a modified set of a complex conjugate, Hermitian conjugates, and bras.
7
Fig. 2. Domain of the delta function
2.3.1. Modified complex conjugate
∗{}
We define a modified complex conjugate for a function of n-parameters f( a ) by
i i=1,...,n
{ }
f({ai}i=1,...,n)∗{ai|i∈A} = f∗({ai}i∈A,{a∗i}i6∈A), (2.21)
where A denotes the set of indices attached to the parameters in which we keep the analytic-
ity, and, ontheright-handside, onf actsonthecoefficients includedinf. Forexample, the
∗
complex conjugates and of a function f(q,p) = aq2+bp2 are f(q,p) q = a q2+b (p )2
q q,p ∗ ∗ ∗ ∗
∗ ∗
and f(q,p) q,p = a q2 + b p2. The analyticity is kept in q, and both q and p, respectively.
∗ ∗ ∗
For simplicity we express the modified complex conjugate as .
∗{}
2.3.2. Modified bras and , and modified Hermitian conjugate
m
h | {}h | †{}
For some state λ with some complex parameter λ, we define a modified bra λ by
m
| i h |
λ λ (2.22)
m ∗
h | ≡ h |
so that it preserves the analyticity in λ. In the special case of λ being real it becomes a usual
bra. In addition we define a slightly generalized modified bra and a modified Hermitian
{}h |
conjugate of a ket, where is a symbolical expression for a set of parameters in which
†{} {}
we keep the analyticity. For example, u,v u = u u = m u , ( u )†u,v = ( u )†u = m u . We
h | h | h | | i | i h |
8
express the Hermitian conjugate of a ket symbolically as ( )†{} = . Also, we write
†{} | i {}h |
the Hermitian conjugate of a bra as ( )†{} = . So for a matrix element we have the
†{} {}h | | i
relation u A v ∗{} = v A† u .
{}h | | i {}h | | i
2.4. Properties of qˆ , pˆ , q and p , and a theorem for matrix elements
new new new new
| i | i
The states q and p are normalized so that they satisfy the following relations,
new new
| i | i
q q = δǫ1(q q), (2.23)
mhnew ′| inew c ′ −
mhnew p′|pinew = δcǫ′1(p′ −p), (2.24)
where ǫ = ~ and ǫ = ~m′ω′ . For sufficiently large mω and small mω the delta
1 mω(1 m′ω′) ′1 1 m′ω′ ′ ′
− mω − mω
functions converge for complex q, q , p, and p satisfying the conditions L(q q ) > 0 and
′ ′ ′
−
L(p p) > 0, where L is given in eq.(2.14). These conditions are satisfied only when q and
′
−
q or p and p are on the same paths respectively. In the following we take mω sufficiently
′ ′
large and mω sufficiently small. Then eqs.(2.23)(2.24) represent the orthogonality relations
′ ′
for q and p , and we have the following relations for complex q and p:
new new
| i | i
dq q q = 1, (2.25)
new m new
| i h |
ZC
dp p p = 1, (2.26)
new m new
| i h |
ZC
∂
pˆ q = i~ q , (2.27)
†new| inew ∂q| inew
~ ∂
qˆ p = p , (2.28)
n†ew| inew i ∂p| inew
1 i
q p = exp pq . (2.29)
mhnew | inew √2π~ ~
(cid:18) (cid:19)
Thus qˆ , pˆ , q and p with complex q and p obey the same relations that qˆ, pˆ,
n†ew †new | inew | inew
q , and p with real q and p satisfy. In the limits of mω and mω 0 δǫ1(q q),
| i | i → ∞ ′ ′ → c ′ −
δǫ′1(p p) and exp ipq in eqs.(2.23)(2.24)(2.29) are well-defined as distributions of the
c ′ ~
−
type , which is introduced in subsection 2.2.
(cid:0) (cid:1)
D
For real q and p, q and p become q and p respectively. Also, for them,
′ ′ ′ new ′ new ′ ′
| i | i | i | i
qˆ and pˆ behave like qˆand pˆrespectively. In relation to the disappearance of the anti-
n†ew †new
Hermitian terms in qˆ and pˆ , we put forward a theorem for matrix elements of the form
n†ew †new
q orp (qˆ ,qˆ ,pˆ ,pˆ ) q orp , where (qˆ ,qˆ ,pˆ ,pˆ )isaTaylor-
mhnew ′ ′|O new n†ew new †new | ′′ ′′inew O new n†ew new †new
expandable function of the four operators qˆ , qˆ , pˆ , and pˆ . We easily see that such
new n†ew new †new
a matrix element can be expressed as the summation of the products of factors made of
q , p, q , p or their differential operators and the distributions δǫ1(q q ), δǫ′1(p p ),
′ ′ ′′ ′′ c ′ − ′′ c ′ − ′′
9
or exp ipq . Then, since we shall extract only analytically weighted results from the
~ ′ ′′
±
matrix element, we do not have to worry about the anti-hermitian terms in qˆ , qˆ , pˆ
(cid:0) (cid:1) new n†ew new
and pˆ , provided that we are satisfied with the result in the distribution sense. So we pose
†new
the following theorem.
Theorem: The matrix element q or p (qˆ ,qˆ ,pˆ ,pˆ ) q or p can be
mhnew ′ ′|O new n†ew new †new | ′′ ′′inew
evaluated as if inside the operator we had the hermiticity conditions qˆ qˆ qˆand
O new ≃ n†ew ≃
pˆ pˆ pˆ for q , q , p, p such that the resulting quantities are well-defined in the
new ≃ †new ≃ ′ ′′ ′ ′′
sense of distribution.
This theorem could help us from running into calculations that are too hard for the use of
our complex coordinate formalism.
2.5. Remarks on the complex coordinate formalism
We have seen that qˆ and pˆ etc. have nice properties, but one might still feel a bit
new new
uneasy about our replacement of qˆ and pˆ with qˆ and pˆ . To accept qˆ and pˆ , it
new new new new
might help slightly to have in mind that operators smooth in qˆ and pˆ like qˆ and pˆ
new new
generically have eigenvalues filling the whole complex plane, while Hermitian operators like
qˆ and pˆ have eigenvalues only along a certain curve, e.g., on the real axis in the complex
plane. For our purpose of having general contours running through eigenvalues we replaced
the special operators qˆ and pˆ by the more generic ones qˆ and pˆ . The philosophy
new new
should be that almost any small disturbance would anyway bring qˆ and pˆ into operators
of the generic type with the whole complex plane as a spectrum. The operators qˆ and
new
pˆ are just concrete examples of such tiny deformation. So we stress that the Hermitian
new
operators as qˆ and pˆ are special by having their eigenvalue spectrum on a curve, e.g., on
the real axis in the complex plane rather than distributed all over it. If we had clung to
the belief in curve-spectra, it would have been embarrassing for our formalism that under
Heisenberg time development one could have feared that, from time to time in our scheme,
the curve-spectra would be transformed into new curve-spectra that might not match at the
free contour choice. Now, however, as already stressed, if we use qˆ and pˆ , from the
new new
beginning we have already gone over to operators with any complex numbers as eigenvalues.
So arbitrary deformation of the contour would a priori cause no problems. Thus we claim
that the contours of integration can be chosen freely at each time t, so that there is no need
for any natural choice, which only has to run from to .
−∞ ∞
We come back to the problem we raised at the beginning of this section: how eq.(2.1) is
expressed for complex q. Now we can express it based on our complex coordinate formalism
10