Decaying states and the rigged Hilbert space formulation of quantum mechanics

103

Physica 124A (1984) 103-114 North-Holland, Amsterdam

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATIONOF QUANTUMMECHANICS A. BOHM Physics Department, The University of Texas at Austin, Austin, Texas 78712 Gamow vectors are defined as generalized eigenvectors (functionals) of the s e l f - a d j o i n t Hamiltonian with complex eigenvalue and exponential time development which are connected with a pair of second sheet S-matrix poles. I am happy to give t h i s t a l k at this conference and in Boulder. connected with the subject of my t a l k for two reasons.

Boulder is

I t was here in Boulder almost exactly 17 years ago that I gave a series of lectures on the RHS formulation of q.m. Around that time several people besides myself--including A. Grossman, Roberts, Antoine--had the idea of using the beautiful mathematical structure of the RHS (which was provided by Gel'fand and his school) fo r a rigorous formulation of the Dirac formalism. You learned this morning from Fishers axiom ~ (cf Appendix A) that the role of rigorous mathematics in physical science is to make sense of h e u r i s t i c ideas, not to assert they are nonsense. The Dirac formalism of the bras and kets is my example of such an h e u r i s t i c idea. Von Neumann's H i l b e r t space formulation asserted that i t is mathematical nonsense. The RHS was the r i g orous mathematics that made sense of i t . But that is not where the role of rigorous mathematics ends. After the new mathematical language has been learned in physics i t is to be used to think and to speak. In that way one w i l l be led to new ideas and deeper i n s i g h t , which could not be attained in the old language. This brings me to the second reason why Boulder is such an appropriate place fo r my t a l k . I t is connected with G. Gamow. The new mathematical language of the RHS contains e n t i t i e s that describe exponentially decaying states which o r i g i n a t e from Gamow. This was not foreseen when the RHS formulation of quantum mechanics was conceived to make the Dirac formalism rigorous. We have called these new e n t i t i e s Gamowvectors. In my present t a l k I shall t r y to give a general idea of the RHS formulation and Gamowvectors, deriving them from the poles of the S-matrix. I shall not go into the mathematical d e t a i l s , which w i l l be discussed in a t a l k tomorrow by M. Gadella. In the same session tomorrow on algebras of H i l b e r t Space unbounded operators, in p a r t i c u l a r in the t a l k by Lassner, you can also learn of other new concepts which came out of this new language and which led to results that could not have been obtained in the conventional language. The rigged H i l b e r t spaceI (RHS) or Gel'fand t r i p l e t is a t r i n i t y of spaces ¢cHc¢

x

Presented at the V l l t h INTERNATIONALCONGRESSON MATHEMATICALPHYSICS, Boulder, Colorado, 1983. 0378-4371/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(I)

104

A. BOHM

Here H is a Hilbert space, { is a space with a nuclear topology (often given by a countable number of scalar products) ~ , which is stronger than the H i l bert space topology ~H' @ is TH-dense in H.

~x is the conjugate space of

i . e . the space of all T~-continuous antilinear functionals F(@), # ~ @ , which we denote by {@IF>. {~IF) is an extension of the scalar product (~,f) to those F~@ x which are not f ~ H x = H. To this t r i p l e t of spaces belongs a t r i p l e t of linear operators for every observable: AcAcA x

(2)

Here A is a T~-continuous operator in adjoint (e.s.a.).

÷ in most cases essentially self-

A is the self-adjoint operator in H and the closure of A,

i t is generally not ZH-continuous.

Ax is a continuous operator in ~x (in gen.

eral not bounded) and the conjugate of the operator A defined by (@[AXlF) = ( A¢IF )

for all

@e @ and

IF ) e @x

(3)

FwE~x is defined to be a generalized eigenvector of the operator A with eigenvalue

~ iff

< A~I F> = <¢I AxI F~> = ~ ~I F~>

for all

(4)

@e@

This is often also just written as

AI F > = ~1 F> = ~1~>

(4')

According to Dirac, 2 f o r an observable H (e.g. the energy operator) there e x i s t s a basis system of eigenkets vector @=

IE+), IE~) such, t h a t

every physical state

, can be expanded with respect to t h i s basis system: I

dE [E+>(+EI~> + In IE~)(+EnI@)

(5)

IE~) represent the bound "states" and IE+) the "scattering states". Under mathematically precisely stated conditions (5) can be proven and is a special case of the Gel'fand-Maurin Nuclear spectral theorem,l The IE~) are proper eigenvectors of H and elements of HIEn + ) : EnIE n) +

(6)

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATION

105

The Dirac kets ]E+)are generalized eivenvectors in the sense of (4), i . e . , HXlE+> = EIE~

over a l l

¢E @

(6D)

holds as an equation f o r f u n c t i o n a l s over a l l @ E ~ . of the a b s o l u t e l y continuous spectrum of H. entially

E in (6D) are elements

The Gamow vectors describing expon-

decaying " s t a t e s " are l i k e the Dirac kets generalized eigenvectors.

But, in d i s t i n c t i o n to the Dirac kets they w i l l turn out to be f u n c t i o n a l s not over the whole space ~ but over roughly h a l f of the space H1z~) = ZRl z~)

over a l l

~E~

N H+

i z R = ER - ~ r .

(6G+)

Here H+ denotes the Hilbert space of Hardy class vectors from above and one has H = H+OH_

(7)

where H_ is the space of Hardy class vectors from below. Over roughly the other half 3 @f3H one can define exponentially growing Gamowvectors P

HIZR+> = (ER + i ~) Iz~+)

over a l l

~E @nil_

(6G_)

The RHS formulation was developed in order to make the Dirac formalism mathematically rigorous, 4 so i t has the nice features of this beautiful formalism and many advantages over the conventional Hilbert space formulation of von Neumann.5 But i t was unknown at that time that the RHS formulation would also settle the controversy about the exponential decay law.

In the conventional

Hilbert space formulation one obtains deviations from the exponential decay law. 6 The exponentially decaying Gamowvectors, representing the resonance per se, were an unexpected additional asset of the RHS formulation of quantum mechanics. We shall now give a physicist's derivation of the decaying state for the mathematically simplest but physically not yet t r i v i a l and t o t a l l y unrealistic case. I f one wants to derive a decaying or resonance state vector, one must f i r s t decide from what one wants to derive i t .

The most popular description of

resonances by a Breit-Wigner amplitude is only an approximation, so one can use i t and conjecture the Gamowvectors from i t , but for a derivation one needs a mathematical definition of a resonance. So the f i r s t question to decide is:

106

A. BOHM

"What is a resonance?" There are various d e s c r i p t i o n s of a resonance or q u a s i s t a t i o n a r y state which are a l l more or less connected with each o t h e r . 7

This is shown in Table I ,

where also the r e l a t i o n s are given which connect one d e s c r i p t i o n with the other: A resonance leads always to a q u a s i s t a t i o n a r y state and vice versa.

A Breit-

Wigner amplitude always implies a resonance or q u a s i s t a t i o n a r y state but a resonance implies a Breit-Wigner amplitude only i f an a d d i t i o n a l c o n d i t i o n (dn/dE n a(E) = small f o r n > 4) is f u l f i l l e d .

A Breit-Wigner amplitude also

does not imply the S-matrix d e s c r i p t i o n without any f u r t h e r c o n d i t i o n s .

But

a p a i r of second sheet poles of the S-matrix w i l l always lead to the three other occurrences of resonances l i s t e d in Table I .

Therefore we shall take f o r the

d e f i n i t i o n of a resonance the p a i r of second sheet poles of the S-matrix at L and ER + i F ER - i ~

Table 1

What is a decaying state? Resonance

(defined by:

t~(E) = 2

r a p i d l y increasing 6c(E))

~z(E) dE

q u a s i s t a t i o n a r y State

(defined by:

F

i26~(E) S~(E) = e

0

large value f o r time delay t~(E))

E

= 1 + 2i

ER -E - i

only i f

dn6(E) = small f o r n > 4 dEn

Breit-Wigner Amplitude

only if

( S i e g e r t pole)

A n a l y t i c i t y (Causality) and U n i t a r i t y

Pair o f second sheet poles o f S-matrix.

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATION

107

Table 2

State:

Stationary

S-matrix description:

Poles on the negative real axis

Vector space description:

Eigenvector of H

Quasistationary

Scattering

Poles in the second sheet

Cut along Spect. H Generalized

~

Eige_nvectors of H

Table I I shows the correspondence between the S-matrix description and the vector space description of the various states.

For stationary states and f o r

scattering states one has an S-matrix description as well as a description by vectors.

For decaying or quasistationary states one had so f a r only the S-mat-

r i x description.

The missing vectors in I

? 1

of Table I I are the Gamow

vectors. The representation of the S-matrix which I s tar t with in order to derive the 7

Gamow vector f o r the S-matrix pole is the following: For a l l t: - ~ < t

+~ one has:

(~°ut(t),s@in(t)

: (R-~°ut(t),~+@in(t)) :

(~-(t),@+(t))

= [

dE (~-IE-) S(E + iO) <+El@+ )

(B)

Sp H @+(t) = e-iHt@+ develops from prepared in-state @in ~-(t) = e-iHt~ IE ± > = IE

÷

develops into measured out-state ~out

1 - --HV ±iO I E > : E

~±IE> (H -V)

IE > = E]E )

<+Elm+>=

El@in>

is given by the incident beam resolution

<-EI~-> :

EI~°ut>

is measured as energy d i s t r i b u t i o n of the detected state

(9)

(I0)

The notation here is standard; 7 (9) is called the Lippman-Schwinger equation, ~+and ~- are the M~ller wave operators.

The complex Riemann surface f o r the

S-matrix has two sheets with a cut from 0 to +~, as shown in Fig. la and Fig. lb.

For the decaying states we need Fig. la which shows the upper h a l f plane

108

A. BOHM

of the f i r s t

sheet (physical sheet) and the lower h a l f plane of the

second sheet.

The i n t e g r a t i o n in (8) over the spectrum of H is along

the upper rim of the cut in the f i r s t

sheet o r , which is the same, along the

lower rim of the cut in the second sheet.

This is i n d i c a t e d by E + iO f o r the

S-matrix in (8) and by S(E + iO) = S(EII - iO), where the subscript I I r e f e r s to the second sheet and by the + preceding E in (+El and the IE->.

We r e s t r i c t

f o l l o w i n g E in

ourselves in t h i s discussion to the simplest n o n - t r i v i a l

case in which the S-matrix element (one p a r t i c u l a r p a r t i a l wave and a l l other quantum numbers f i x e d and ignored) S(w), where ~ is the complex energy v a r i a b l e , has no other s i n g u l a r i t i e s but one p a i r of resonance poles at ~ = z R = ER - i ~r (r > O) and ~ = Z~ in the second sheet.

And the spectrum of H is the

p o s i t i v e real l i n e (there are no bound state or v i r t u a l state poles). 8

E [first sheet)

E($econd shee~) (b) E ( second sheet)

/ / / / / /

Figure I .

F[flrst sheet]

Deformation of the path of i n t e g r a t i o n i n t o the second sheet of the energy plane. Part (a) is f o r the decaying s t a t e , (b) f o r the growing s t a t e .

Now we shall deform the contour of i n t e g r a t i o n i n t o the second sheet o f the lower h a l f plane through the cut, as shown in Fig. l a .

Then we obtain an inte-

gral along the contour C below the pole plus an i n t e g r a l around the resonance pole at z R

(~-'@+) : I

d ~ < @ - I m - ) S l l ( m ) (+~*l@+)

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATION

C

109

R

s_1 where S_l is residium of S(w) at ZR, i. e. , SII(m) : m _ZR + s O + S l ( ~ - ZR) .... The second term can be evaluated using the Chauchytheorem to gJve the pole term (@-]ZR-) (-2~is_l) (+ZRl@+>

(12)

Both contours can be further deformed i f the functions under the integral f u l fill

certain conditions, roughly i f they are analytic functions vanishing at

the lower i n f i n i t e semicircle in the second sheet s u f f i c i e n t l y f a s t .

This i s

connected with 4- being an element of H+ ( < -El4-) being a Hardy class function from above and i t s complex conjugate (4-1E-) being a Hardy class function from below). ( I I ) then takes the form

(~b-,¢+) = I ~°II

dE <~-IEII> SII(E) <+EI~+>

0 + i~

dE<~-IE-) ~S_l

<+Elq~+>

-~II I f ~- E ~C4

II (13)

<~-IZR-) (-2~is_l) <+ZRl@+)

The pole term (12) has now become a generalized Breit-Wigner integral; the integration extends from -~II to +~ in contrast to the conventional BreitWigner integrals in which the integration extends from 0 to +~. The equality of the pole term (12) and the generalized Breit-Wigner integral in (13), f~

S_l dE<~-IE-) E---~R <+EI¢+): <~-IZR-)(-2~is-l)<+ZR I@+>

'

(13a)

is a consequenceof the Titchmarsh theorem for Hardy class functions (4-1E-) from below (i.e. for 4- ~ll+). The f i r s t integral (background term) can also be rewritten a l i t t l e bit i f one defines in analogy to the usual relation

110

A. BOHM

I E+ > = I E- > S(E)

(14)

an I EII+) also on the lower rim of the negative real axis of the second sheet IEII+ ) = IEII- ) SII(E)

(1411)

I f one then omits the a r b i t r a r y ~-E~CIH+ in equation (13), one can w r i t e for ~ + ( ~ CIH+ a new basis system expansion

dE IE+><+EIm+> + Iz R-

=

) (-2~is_l)

<+ZRl~+>

(15)

0 where IZR-) is defined according to (13a) by I f+~ IZR- ) = -2~-T

I dE [E-) E -z R

(16)

-~II (E in (16) is along the lower rim of the real axis in the second sheet). In distinction to the Dirac basis system expansion (5), whose generalized basis vectors can be considered as functionals over the whole space ~CH+(~)H_, the generalized basis vectors in (15) are to be considered as functionals over the {~-} C~ NH+.

For ~-(~H+ i.e. when (~-I E-)

are Hardy class func-

tions from below, the definition (16) is the Titchmarsh theorem. IZR-) is a generalized eigenvector of H, because ~- is also ~ @ which means that also
replaced by

to give

I I~ E : ZR[ZR > HIZR-) : -2-~Tj dE I E-) E-=R -~II

(17)

This equation is true only i f IZR-> is considered as functional over the ~ - ~ C I H + , in which case (17) is again the Titchmarsh theorem for Hardy class functions from below E(~-IE- ) . (17) is the desired property (6G+) and (16) defines--except for "normalization"--the exponentially decaying Gamowvector. I t is a functional over {~-}. The exponentially growing Gamowvector, defined by *+ 1 I=R > : ~

[4~ .

-~II

dE [

E+

) ~1

. ,

,

?

z R : ER + i ~

(18)

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATION

111

(E in (18) is along the upper rim of the real axis in the second sheet) considered as functional over @+(~@NH

is obtained in an analogous manner: starting

from the complex conjugate of the S-matrix representation in (8) one deforms the contour into the upper half of the second sheet. pole in the second sheet above the real axis.

(18) then comes from the

For the Iz~+) one proves (6G_).

We w i l l now show that the exponentially decaying Gamowvectors (16) decrease exponentially in time.

Let ~- E{@-}CH+ i . e . (~-IE-)

tion from below. Then "e-iHt"IZR- )

is a Hardy class func-

can be defined by

<~-i"e-iHt"iz R- ) ~ = -2-~31 rj +~ dE <~-IE-> e-iEt

l (19) E -z R •

-~II I f <~-IE-)

is a Hardy class function from below then

(~-IE-) e"iEt

is also a

Hardy class function from below as long as t ~0 but not for t < O. Therefore for t ~ 0 one can choose for 4- in the above derivation 4- = eiHt~-~@ ~ H÷ and ~Ht" zR- ) can be defined. The Titchmarsh theorem applied to the r.h.s, of (19) then gives -iz Rt (@-I"e-iHt"IZR- ) = e <~-IZR-)

for

t ~ 0

(20)

Therewith we have seen that time development can be defined for the IzR->but only i f t ~0.

Then

_iHtlz -iERt -(P/2)t e R ) =e e IZR->

for t ~ 0

considered as a functional over @C ~ H + R Th.us ing Gamowvector.

(21)

Iz ")

is an exponentially decay-

Time development "e-iHt"cannot be defined for t < 0 on IZR->

or any functional over @NH+. By analogous arguments one shows that the Iz~+)

of (18) are exponentially

growing Gamowvectors but only for time t <_0 " e-iHt" iz +> : e-iERte+(P/2)t Iz *+) R

for

t _< 0

(22)

With the result (15) and (21) we can now address the question which was the incentive for this investigation: Are decaying states exponentially decaying? IZR") (pole term) describing the resonance per se has exponential time development (but only into the forward direction t > O) But IZR-) is not element of the space ~ .

The "physical" state @+ in (15)

1 12

A. BOHM

has always a background integral describing the effect of the preparation. This background does not have exponential time development, for a suitable choice of the space ~ i t can be arbitrary small but not zero. I f per definition only the ~+ are physical states, then there are no expone n t i a l l y decaying physical states.

But i f also IZR-)

state, then this state is exponentially decaying.

can describe a physical

Which of these alternatives

to choose is certainly not a question that one can decide by mathematics, as has been tried in the past. Summarizing the contents of this lecture and other results which have not been mentioned here, we want to answer the question:

What do Gamowstates do

for us? l)

fDecaying state\ The ~Growing state J is connected with second sheet pole of S-matrix

DECAYING STATES AND THE RIGGED HILBERT SPACE FORMULATION

old:

@+ : [ "~ dE IE+)(+EI(h +) 0

new:

@+ = I - ~ d E ,EII+)<+EII , I@+)+ IZR") 0

113

(+ZRl¢+)

REFERENCES I ) I M Gel'fand and G P Shilov, Generalized Functions, VoI. 4 (Academic, New York, 1964); K Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups (Polish S c i e n t i f i c Publishers, Warszawa, 1968); A Bohm, The rigged Hilbert space and quantum mechanics,in: Springer Lecture Notes in Physics, Vol. 78 (Springer, New York, 1978). 2) P A M Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1958). 3) For the precise mathematical d e f i n i t i o n of the spaces over which the Gamow vectors are defined see the paper by M Gadella presented at this meeting.

4) J E Roberts, J. Math. Phys. 7, 1097 (1966); Commun. Math. Phys. 3, 98 (1966); A Bohm, "The Rigged Hilbert Space in Quantum Physics, ICTP Report No. 4, Trieste (1965); A Bohm, in: Boulder Lectures in Theoretical Physics, 1966, Vol. 9A (Gordon and Breach, New York, 1967), p. 255; J P Antoine, J. Math. Phys. I0, 53 (1969); I0, 2276 (1968); 0 Melsheimer, J. Math. Phys. 15 (1971) 902. 5) J von Neumann, Mathematical Foundations of Quantum Mechanics (Springer, New York, 1932); G Ludwig, Grundlagen der Quantenmechanik (Springer, Berlin, 1954). 6) L Fonda, G C Ghirandi, A Rimini, Decay theory of unstable quantum systems, Reports on Progress in Physics, 41, 587 (1978). 7) A Bohm, Quantum Mechanics (Springer, New York, 1979), Chapters XVlII and XV. 8) The extension of this formalism to the case of a countable number of resonance poles is straightforward; instead of one Gamowvector one obtains a sum over the countable number of Gamow vectors. Bound state poles just add a discrete sum to the spectral resolution. Cuts in the complex plane w i l l a l t e r the background integrals in (13) below. Virtual state poles can also be included, see references 3 and 9. 9) M Gadella, On the RHS Description of Resonances and ~irtual State Poles, University of Santander, Spain preprint (1983). APPENDIX From t a l k "What's Mathematical Physics to Physicists?

Some examples from

Past, Present and Future" by M Fisher, V l l t h International Congress on Mathematical Physics, Boulder, Colorado, August 1983. Problems: Some Observations

1 14

A. BOHM

Axiom ~ " C o m p r e h e n s i b i l i t y " "Theorems incomprehensible to a t h e o r e t i c a l p h y s i c i s t are not theorems in MA@."

[Proofs need not be i n t e l l i g i b l e :

Superaxiom ~ " I n t e l l e c t u a l

M ~ @]

Life"

"Fields of study . . . George A. Keyworth I I " I t is d i f f i c u l t

to round up . . , money to study questions few

people can a p p r e c i a t e . " Axiom ~

"Positivity"

"A r o l e of rigorous mathematics in physical science is to make sense of h e u r i s t i c ideas ( i . e . -

f i n d the " c o r r e c t s e t t i n g " )

not to assert they are nonsense" Caution re "No Go theorems"