The Rigged Hilbert Space Formulation of Quantum Mechanics An

7/22/2019 The Rigged Hilbert Space Formulation of Quantum Mechanics An

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arXiv:quant-ph/9505004v1

12May1995

ASI-TPA/6/95

The Rigged Hilbert Space Formulation ofQuantum Mechanics and its Implicationsfor Irreversibility

Christoph Schulte, Reidun Twarock

Institut fur Theoretische Physik (A)TU Clausthal

38678 Clausthal-Zellerfeld, Germany

based on lectures by

Arno Bohm

CPP, Physics DepartmentUniversity of Texas at Austin

Austin, TX 78764, USA

Abstract

Quantum mechanics in the Rigged Hilbert Space formulation describes qua-sistationary phenomena mathematically rigorously in terms of Gamow vectors.We show that these vectors exhibit microphysical irreversibility, related to anintrinsic quantum mechanical arrow of time, which states that preparation ofa state has to precede the registration of an observable in this state. More-over, the Rigged Hilbert Space formalism allows the derivation of an exact

golden rule describing the transition of a pure Gamow state into a mixture ofinteraction-free decay products.


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Introduction

The idea of using the Rigged Hilbert Space formulation to describe quantum me-chanics already occurred in 1966 in [1] and was later elaborated in [2] and [3]. Thisformalism recaptures all standard results of quantum mechanics and makes the math-ematical theory of Diracs bras and kets rigorous. Unexpectedly it was found [3, 4]that it also allows the description of resonances and decaying states. Resonancesin scattering theory are usually defined as the poles of the analytically continuedS-matrix. The Rigged Hilbert Space provides a means to describe these states bymeans of vectors. We call these vectors Gamow vectors. Their time evolution isgiven by a semigroup instead of the usual unitary one-parameter group [4,5]. Thisparticular feature made people aware of a connection between Gamow vectors andirreversibility. For example Antoniou, Prigogine et. al. who tried to obtain intrinsicirreversibility on the quantum level [6] advocated the use of this method to explainirreversibility on the microphysical level [7]. It finally could be shown in [8] that the

occurrence of the two semigroups is linked to a quantum mechanical arrow of timewhich stems from the fact, that states must be prepared before observables can bemeasured in them.

In section1we briefly describe the Rigged Hilbert Space formalism of quantummechanics. A more detailed presentation is provided by the above references, e. g.[5].

Section2is devoted to the derivation of a quantum mechanical arrow of time fromthe time shift between the preparation and registration procedure. This quantummechanical arrow of time is based on an earlier arrow of time formulated by Ludwigand based on the Bohr-Ludwig interpretation of quantum mechanics [9] which howeverwas only formulated for the experimental preparation and registration apparatuses

and not incorporated into quantum mechanics. The reason for this was, that thestandard quantum mechanics in Hilbert space is reversible and cannot accommodatean arrow of time.

In section 3.1 a mathematical formulation of the quantum mechanical arrow oftime for general scattering experiments is given. We will cast it in a form whichreappears in section3.2, where the quantum mechanical arrow of time in connectionwith Gamow vectors is discussed. Finally, we consider decaying states and show howthese results lead to an exact golden rule.

1 The Rigged Hilbert Space Formalism of Quan-

tum Mechanics

By considering different topologies on a given linear space, we can produce variouscomplete topological spaces. This idea is behind the Rigged Hilbert Space (R.H.S.)


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formalism, which uses Gelfand triplets of the form

H , (1)

where

Hrepresents a Hilbert space with the well-known Hilbert space topology H

and a subspace with a nuclear topology . This topology is chosen to be strongerthan the Hilbert space topology.

Furthermore, is the conjugate space of , i. e. the space of all -continuousantilinear functionals F(), , in the following denoted by F() =< |F >.Notice, that < |F > with F is an extension of the ordinary Hilbert spacescalar product (, f) for f H=H.

The construction of the above triplet is such that is H- dense inH. The space will be used later to define the physical states.

In order to describe observables, we need the following triplet of linear operators

A

|

A

A, (2)

where A has to be a -continuous operator in but is not a closed operator inH.The operatorA is the Hilbert space adjoint and Athe closure ofA. In generalAandA are not H-continuous. A

, defined by

< |A|F >=< A|F >for all and|F > (3)

is a continuous operator in , called the conjugate operator of A. Thus A isdefined as the extension ofA to . In this formalism, observables are representedby elements of an algebra of -continuous operators whereas in the usual Hilbertspace formulation they are given by linear (unbounded) operators defined on

H. In

the Hilbert space formulation, of quantum mechanics (pure) states are described byelements ofH; in the Rigged Hilbert Space formulation only elements of representphysically preparable states.

Generalized eigenvectorsF of the operator A with eigenvalue are defined by

< A|F >< |A|F >= < |F > for all , F . (4)

The Dirac kets are generalized eigenvectors in this sense and the expansion of aphysical state vector for an observable Hin terms of a basis system of eigenkets, isgiven by the generalized basis vector expansion which Dirac already used but whichwas proven only by the Nuclear Spectral Theorem [10]

=0

dE|E >< E| >+ En

|En)(En|) for every , (5)

whereH|En) =En|En), H|E >=E|E > (6)


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The physical interpretation of these two triplets will be the following: is the spaceof the preparations, i. e. the physical in-states, + is the space of the registrations,i. e. the detected out-states of the scattering experiment.

It can be shown, that the restriction of the unitary time evolution operatorU

(t) =

eiHt

to is a -continuous operator with

U(t)+ + only for

t 0and analogously a -continuous operator with U(t) only for t 0. This

results in a pair of evolution semigroups instead of the ordinary unitary evolutiongroup U(t) which is the mathematical manifestation of irreversibility.

2 Quantum Mechanics and the Arrow of Time

In the conventional description of quantum mechanics irreversibility is not accountedfor intrinsically and the time evolution is given by the unitary one-parameter group ofoperators U(t) =eiHt which is generated by the Hamiltonian Hand well-defined for

< t


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the observational facts into the mathematical theory. This was necessary becausehis Hilbert space formulation of quantum mechanics does not allow for an intrinsicquantum mechanical arrow of time according to the Bohr-Ludwig interpretation ofquantum mechanics. The division of an experiment into a preparation apparatus,

which prepares physical states for the experiment, and a registration apparatus, whichmeasures properties of the microsystem, is crucial. The microsystem itself is the agentby which the preparation apparatus acts on the registration apparatus. Whereas thestate was prepared by the preparation apparatus the measurements of the observablesF is done by the registration apparatus and the expectation value represents theaverage value of an ensemble of measurements:

Tr(W F) =

( , F ) =

< | >< | > . (13)

Here, we choose the special case that the statistical operator Wis given by the vector describing a pure state and the self-adjoint operator F is the projection operator

| >< | describing the special observable called decision observable or property.Whereas the apparatuses are not described by quantum mechanics, the measured

value obtained by the measuring apparatus is predicted by quantum theory. Weobserve the dynamics of the microsystem as the time translation of the registra-tion procedure relative to the preparation procedure. Thus, if a time direction isdistinguished for the time translation of the registration apparatus relative to thepreparation apparatus then this is a distinguished direction for the time evolution ofthe microphysical system described by quantum mechanics. In terms of the apparatusit is possible to describe the arrow of time independently of a mathematical theory.We call this arrow the preparation registration arrow and characterize it by thefollowing statement [9]:

Time translation of the registration apparatus relative to the preparationapparatus makes sense only by an amount 0.

Translated into the quantum mechanical notions of the mathematical theory, thisarrow is formulated as:

An observable|() >< ()| in particular a projection operator can be measured in a state = (0) only after the state has been prepared,i. e. for 0.

This is the formulation in the Heisenberg picture, in the Schrodinger picture it isgiven as:

The state (t) must be prepared before an observable | > < | =|(0)>< (0)| can be measured in that state, i. e. at t 0.


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The latter two formulations of the state observable arrow of time concern thetheoretical description of our arrow of time; we call it the quantum mechanical arrowof time. These formulations should be equivalent to the first version which ratherconcerns the experimental implications of the quantum mechanical arrow of time.

All version are general expressions of causality and none of them has anything todo with the change of the state due to measurement, i. e. the collapse of the wavefunction.

In the next section we shall apply this quantum mechanical arrow of time to aresonance scattering process. Before we can give a mathematical formulation for thisquantum mechanical arrow of time we will have to introduce the standard formalismof scattering theory and refine it to the Rigged Hilbert Space formulation of quantummechanics.

3 Mathematical Formulation of the Quantum Me-

chanical Arrow of Time in a Resonance Scatter-ing Experiment

3.1 Resonances of the S-Matrix and Gamow Vectors

The design of a scattering experiment is the following: A mixture of initial statesin

is prepared and evolves according to the free Hamiltonian K

in(t) =eiKt/hin. (14)

The evolution through the interaction region is given by the HamiltonianH=V+ K.

The result is a state out,

out(t) =Sin(t), S= + (15)

where

+in(t) +(t) =eiHt/h+ = out(t) (16)out(t) (t) =eiHt/h, (17)

and + and are the Mller wave operators. The state vector in is determined bythe preparation apparatus. The state vector out is determined by the preparation

apparatus and

the dynamics. The detector does not register out

, but a property|out >< out|, where out is determined by the registration apparatus. The prob-ability to measure the observable|out >< out| in the state in is then given by|< in|out >|2 which is the modulus square of the S-matrix.

The entries of the S-matrix are given by the elements

out(t), out(t)

=

out(t), Sin(t)

, (18)


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(H)

E (first sheet)

E (second sheet)

zR

C+

E (second sheet)

E (first sheet)

zR

(H)

C

Figure 1: Paths of integration

which, for allt, can be given by

(out(t), Sin(t)) = (out(t), +in(t))

= ((t), +(t))

=(H) dE <

|E > SI(E+ i0)< +E|+ > .(19)

Here SI(E+ i0) indicates integration along the upper rim of the cut along the realaxis in the first sheet. Since SI(E+i0) = SII(E i0) this integration along thatsheet can be realized by the integration over the lower rim of the cut in the secondsheet. The + precedingE in < +E| and the following E in|E > underline thisfact.

In order to define now the Gamow vector in terms of the S-matrix we consider thesimplest model and we make the following assumptions (if there are more resonance


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poles and also bound states the arguments are easily generalized):

There are no bound or virtual states. The spectrum (H) of H = K+V is given by the positive real axis which

implies that there is a cut from 0 toon the Riemann surface for the analyticcontinuation S(z) of the S(E)-matrix .

There is a pair of poles on the second sheet of the complex energy surface ofthe S-matrix S(z) at the pole position:

zR = ER i2

, zR= ER+ i

2 . (20)

Physically, the poles in the second sheet of the S-matrix are associated with reso-nances. We will associate to these pole positions autonomous resonance states and

we call the corresponding generalized state vectors the Gamow vectors. To accomplishthis we deform the contour of integration in (19) into the second sheet and obtain

(, +) =

Cdz < | > SII(z)< +z|+ >

+

dz < |z > s1zzR < +z|+ >(21)

where s1 = i is the residuum of SII(z) at zR and C is the curve in the lowersecond sheet which is shown in figure1. We also have assumed here, that < |z >=< z| > and < +z|+ > are analytic functions in the lower half plane of the secondsheet of the Riemann surface. As the first integral is not related to the resonance wewill ignore this term in the following.

The second integral in (21) can now be evaluated by means of the Cauchy Theo-rem, which yields

< |z+R >(2)< +zR|+ >=

dz < |z >< +z|+ > iz zR . (22)

We now demand further, that the second integral of (21) is given by a Breit-Wignerintegral, i. e. that (22) takes the form

< |zR >< +zR|+ >2 =

+II

dE <

|E >< +E

|+ > i

E(ERi

2 )

.(23)

The reason for this requirement is that we want to associate the resonance statevector with the pole of the S-matrix and the resonance state vector should have aBreit-Wigner energy distribution.


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In order to deform the contour of integration from the one in ( 22) into an integra-tion from to + as in (23), we need to make the following requirement of theenergy wave functions:

< E

|out >=< E

| >

S H2+ or <

|E >

S H2 (24)

< E|in >=< +E|+ > S H2 or < +|E+ > S H2+ (25)

because then < |E >< +E|+ > S H2 and (23) is a special case of theTitchmarsh theorem (cf. Appendix). In hereH2+ means Hardy class from above andH2 means Hardy class from below.

If the conditions (24) and (25) are fulfilled, we say that is a very well-behavedvector from above and + a very well-behaved vector from below and write thisas

+, + . (26)

This means the space of very well behaved vectors from below, , describes the statevectors prepared by the preparation apparatus, and the space of very well behavedvectors from above, +, describes the observables detected by the registration appa-ratus. We know that a more careful investigation would show that + ={0}(zero vector) [5]. We shall see below, that conditions (24) and (25) are exactly themathematical statement describing the quantum mechanical arrow of time. It willthus come as no surprise that the time development of the Gamow vectors is givenby two semigroups instead of a unitary evolution group.

It may seems curious that we label the vectors , + by superscripts that areopposite to the subscripts by which we label their spaces +, . The reason for thisis that the labelling of the vectors comes from the convention of scattering theory

developed by physicists and the labelling of the spaces comes from the conventionfor the Hardy class spacesH2+ andH2 used by mathematicians. Both subjects weredeveloped independently and without knowing of each other. The agreement between{} and +as well as {+} and is an example of the unreasonable effectivenessof mathematics in the natural sciences (E. P. Wigner).

We are now able to define the Gamow vectors|zR >and|zR+ >associated withthe poles at zR and z

R. In analogy to (23) we apply the Titchmarsh Theorem to the

function< |E > H2 with value < |zR >at zR to obtain:

< |zR >= 12i+II

dE < |E > 1EzR

for all < E| > S H2+(27)

and to the function < +|E+ > H2+ with value < +|zR+ > at zR to get:

< +|zR+ >= 12i+II

dE < +|E+ > 1Ez

R

for all < +E|+ > S H2.(28)


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Notice, that in equation(27), integration is taken along the lower rim on the secondsheet whereas in equation (28) along the upper rim on the second sheet.

It is clear that|zR >defined by (27) is an element of + and|zR+ >defined by(28) is an element of . These generalized vectors (functionals) are the two kinds of

Gamow vectors. It is crucial that the integration in (27) and (28) extends fromto +though the spectrum ofHis bounded from below.Replacing by H (respectively + by H+) in equation (27) (respectively

equation (28)) and using< H|E >=E < |E > (respectively< H+|E+ >=E < +|E+ >) we can infer that the Gamow vector|zR > (respectively|zR+ >) isa generalized eigenvector ofHwith the eigenvalue zR (respectively z

R):

< H|zR >< |H|zR >=zR < |zR > (29)or H|zR >=zR|zR >;|zR > +

< H+

|zR >

< +

|H

|zR

+ >=zR < +

|zR

+ > (30)

or H|zR+ >=zR|zR+ >;|zR+ > .

3.2 Time Evolution of the Gamow Vectors

In the usual Hilbert spaceH, the time evolution, given by eiHt , is a continuousoperator for all tR. The time evolution operators of the Gamow vectors|zR >+ and|zR+ > are the conjugate operators ofU(t)|+ =eiHt |+ and U(t)| =eiHt |, respectively. The conjugate operators can only be defined if the operatorsthemselves are continuous operators with respect to the topology in . In the caseofU(t)

|+ this is only the case for t

0 and we denote the conjugate by e+

iHt.

On the other hand the operator U(t)| is a continuous operator in only for t 0and here, the conjugate is denoted by e

iHt. Thus, eiHt+ exists only fort0and eiHt exists only fort 0.

Now, the time evolution of the Gamow vectors|zR >+ can be calculated tobe

eiHt|zR

e+iHt zR

= eiERte

2t |zR for all +, and for t 0

(31)

and the time evolution of|zR+ > is calculated to be:

eiHt+|zR+ + eiHt zR+

= eiERte

2t

+|zR+

for all + , and for t0 .(32)

Thus the vector|zR > is an exponentially decaying state vector. If< |zR >

2 isthe probability of finding the Gamow state|zR >with the detector that registers the


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property| >< | at t= 0 then< |eiHt |zR >

2 =et< |zR >

2 (33)is the probability to find this Gamow state at the time t. Thus the probabilitydecreases exponentially towards 0 as tgrows. Analogously, the increasing state |zR+ >is defined for t 0 with probability increasing exponentially from 0 in the distantpast to

< +|zR+ >2 at t = 0.

This behaviour can be summarized by saying that|zR > and|zR+ > have anintrinsic arrow of time which evolves only from 0 toor fromto 0, i. e.|zR >does not grow and|zR+ >does not decay.

Thus a microphysical system, described by such a state vector, is equipped withan arrow of time.

3.3 A New Generalized Basis Vector ExpansionThe basis vector expansions, of which the Nuclear Spectral Theorem (5) is an example,provide very important tools for quantum mechanics. They are generalizations of thewell known basis vector expansion X=

3i=1 x

iei in the three dimensional EuclideanspaceR3. In theN-dimensional complex space this expansion in terms of eigenvectorsei,i = 1, . . . , N of any self-adjoint linear operator His called the fundamental theoremof the linear algebra. In the infinite dimensional Hilbert spaceH this expansion

H h=n=1

b

|En, b)(b, En|h);|En, b) =ei H (34)

is only correct for a subset of self-adjoint operators H(compact). Here we have calledbthe degeneracy index: H|Enj , b) =En|Enj , b). For any arbitrary self-adjoint operatora generalization of (34) holds, which is given by the fundamental Nuclear SpectralTheorem of the generalized Dirac basis vector expansion (5). (To take degeneracyinto account the label b should be added:|E,b >and a sum should be taken over allvalues of the quantum numbersb.) The set of generalized eigenvalues {En, E} is calledthe spectrum (we have assumed that the spectrum of H was absolutely continuousand discrete).

With the help of the Gamow vectors we can generalize the Dirac basis vectorexpansion further. For a self-adjoint Hamiltonian H = K+V for which there are

onlyNsimple poles of theS-matrix atzRi =Ei ii2, i = 1, . . . , N one obtains (from

a generalization of equation (21) from 1 to Npoles) a new generalized basis vectorexpansion

+ = N

i=1

b |zRi, b >(2i)< +b, zRi |+ >

+0

b dE|E, b + >< +b, E|+ > for every + .

(35)


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We have ignored the sum over the discrete spectrum

n=1 |En)(En|+), correspondingto bound states.

In contrast to Diracs basis vector expansion (5) which holds for every , thenew basis vector expansion (35) holds only for+ = + + and contains

generalized eigenvectors|zRi

>

+. It means that a vector+

representing a state(ensemble) prepared by a preparation apparatus of a scattering experiment can beexpanded into a superposition of Gamow vectors representing exponentially decayingresonance states anda background term

+bg= 0

b

dE|E, b + >< +b, E|+ >; < +b, E|+ > H2 S. (36)

The integration in the background term is taken in the second sheet along the negativereal axis (and the values of < +E|+ > can be calculated from the physical values< +E|+ >=< E|in > with 0 E < + given by the energy distribution of theincoming beam using the von Winter Theorem [5]) and could have been replacedby integration along other equivalent contours. The result (35) shows that resonancestates can take a very similar position as bound states in the basis vector expansionof states + prepared by a scattering experiment. These state vectors +can indeed be given by a superposition of exponentially decaying resonance statesDi =|zRi >

2i (except for a background term which one would try to make

as small as possible), as used so often in phenomenological discussions of physicalproblems.

3.4 The Quantum Mechanical Arrow of Time and ResonanceScattering

We now want to discuss the connection between the Gamow vectors intrinsic arrow oftime and the more general quantum mechanical arrow of time that we had introducedin section 2 and which was nothing else but a general expression of causality. Forthis purpose, we apply that general arrow of section 2 to the resonance scatteringexperiment.

We choose as t = 0 the point in time at which the preparation of the state in(t)is completed and after which the registration of|out >< out| begins. Since nopreparations take place for t > 0, the energy distribution given by the preparationapparatus must vanish and we require therefore that < E|in(t) >= 0 for t >0 andfor all physical values ofE. And since as a general result of scattering theory

< E|in >=< +E|+ > and < E|out >=< E| >, (37)where|E > and|E >are defined by the Lippman-Schwinger equation

|E >=|E >+ 1E H i0 V|E >=

+|E > (38)


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and by K|E >= E|E > and H|E >= E|E >, we formulate this requirement ofthe quantum mechanical arrow of time as:

0 =

dE = dE

=

dE eiEt/h

F(t) for t >0.(39)

Notice thatF is the Fourier transform of the energy wave function =< E|in >.

Similarly, since all registrations take place after t = 0,

0 =

dE < E|out(t)>=

dE eiEt/h G(t) for t 0 and of noregistrations fort as well as < E| > are the energy distributions of theexperimental apparatuses, they are supposed to be smooth, well-behaved functionsof E. Thus, the Theorem of Paley-Wiener is applicable, which says that a squareintegrable function G+(E) (respectively G(E)) belongs toH2+ (respectivelyH2) ifand only if it is the Fourier transform of a square integrable function which vanisheson the interval (0, ) (respectively (, 0)). In our situation this yields

< E| > H2+ S (or +) (41)< +E|+ > H2 S (or + ). (42)

These are precisely the conditions (24) and (25) which we obtained above from therequirement that the pole term of the S- matrix has a Breit-Wigner energy distribu-tion. This means that we could have taken the quantum mechanical arrow of timein its mathematical formulation (39) and (40) as the starting point and derived fromit the Rigged Hilbert Spaces + H + of observables (or out-states ) and H of states (or in-states+) and their Gamow vectors|zR >and zR+ >respectively.

4 Exact Golden Rule

Decaying states can be considered as resonances for which the production process isignored. As an example one can think of the radiative transitions of excited atoms


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A to lower states AbyA A +. Whereas stationary states are rare in physics,decaying states are numerous and thus constitute an important part of physics.

If the decaying system is isolated, its state W(t) evolves in time according to theexact Hamiltonian Hand is given by

W(t) =eiHtW(0)eiHt , with H=K+ V 0. (43)As registration apparatus we choose detectors which surround A. We suppose thatthe decay products are far enough from each other so that they do not interact afterthe decay. Then the observables are projection operators (or positive operators)on the space of physical states of the decay products and are given by

=0

dEb

|E, b >< E,b| (44)

where|E,b >are eigenvectors ofKwith eigenvalue Eand not eigenvectors ofH.

Denote byP(t) the transition probability, i. e. the probability to find the decayproduct in the state W(t). ThenP(t) is given byP(t) = Tr(W(t)). (45)

Inserting for W(t) =|t >< t| with t H one can show thatP(t) is identicallyzero for all tif it was so for some time interval in the past [13]. For the derivative ofP(t) att = 0, i. e. for the initial decay rate P(t= 0) however, the standard treatment[14] leads to an approximate formula given by the Golden Rule, which shows thatP(t = 0) = 0 as it should be. Since nothing more than the most fundamentalassumptions of Hilbert space quantum mechanics enter in this derivation we have toconclude that the transition probability cannot be derived in the framework of theHilbert space formulation in a consistent way.

In contrast to this, in the R.H.S.-formulation one can derive an exact Golden Rulefor P(t) and obtain P(t) as a derivative of it if one chooses for the decaying stateW(t)the Gamow state given by the vector G =|zR > f (with fbeing a normalizationconstant):

W(t) = |G(t)>< G(t)|= et|G >< G| for t 0 only. (46)Then one obtains

P(t) = 1

et

0

dE b=bG

|< E, b|V|G >|2

(E ER)2 + 22 , for t

0, (47)

and from this by differentiation:

P(t) =et20

dEb=bG

2

< E, b|V|G >2

(E ER)2 +2

2 for t 0. (48)


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From this one finds using the fact that the probability to find the decay productat time t 0 must be 0,P(0) = 0 thatP(t) = 1 et and P(0) = (i. e.the initial decay rate is equal to the imaginary part of the resonance pole position ofthe S-matrix, which had already been shown above in section3.1to be equal to the

width of the Breit-Wigner energy distribution).In the Born approximation, which is defined by

G fd, EREd, 2ER

0, (49)

where Kfd = Edfd is the eigenvector of the free Hamiltonian approximating the

decaying state G, one obtains

P(0) = = 20

dEb=bG

< E, b|V|fd >2 (E Ed) . (50)

This is the standard Golden Rule for the transition from the excited but non-interactingstate fd into the mixture of non-interacting decay products.

5 Summary and Conclusions

Quasistationary microphysical systems and their resonances can be described byGamow vectors which are generalized eigenvectors in a suitably chosen space of self-adjoint Hamiltonians with complex eigenvalues. The description of Gamow vectorsis not possible in the Hilbert space, but the Rigged Hilbert Space allows to defineGamow vectors in . These vectors are associated to the resonance poles of the

S-matrix.The time evolution of the Gamow vectors is governed by semigroups of operators

in the spaces and thus displays irreversibility. This is a microphysical arrow oftime which is built in the microphysical systems described by the Gamow vectors.The conjugate semigroups of operators in describe a general quantum mechanicalarrow of time which is the theoretical description of the preparation registrationarrow of time of the experimental apparatuses.

This preparation registration arrow of time has been known for some time,but could not be transcribed faithfully from the experimental observation into themathematical theory of quantum mechanics in Hilbert space. Finally, with theGamow vector to describe a decaying state, one can derive a sensible result for thetransition probability into non-interacting decay products and obtain the standardGolden Rule for the transition rate as the Born approximation. The Gamow vectorprovides the link that was missing from the Hilbert space formulation to connect thetheoretically and experimentally defined quantities for the decay phenomena.


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Appendix

Theorem 1 (Titchmarsh) LetG(z) Hp. Then, for anyz=E0+ iy withy>< +E|+ >.

Similarly, we obtain

< +|zR+ >< zR| >=12i

+II

dE < +|E+ >< E| > 1E(ERi2 )

, (53)

for zR = ER+ i2 and the function G+(E) =< +|E+ >< E| >. Notice, thathere the integration takes place along the upper edge of the real axis in the secondsheet.

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[2] Bohm, A.: The Rigged Hilbert Space and Quantum Mechanics, Lecture Notes inPhysicsVol. 78, 1978.

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[4] Bohm, A.: J. Math. Phys. 221 (1981) 2813; Lectures of Math. Phys. 3, 445(1979).

[5] Bohm, A., Gadella, M.: Dirac Kets, Gamow Vectors and Gelfand Triplets, Lec-ture Notes in PhysicsVol. 348, 1989.

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Prigogine, I.: Exploring Complexity, Freeman, New York, 1988; Antoniou, I:Nature338 210 (1989); Antoniou, I: Proceedings of the II. International WignerSymposium, Goslar 1991, Springer Lecture Notes in Physics, Vol. 379 (1991).

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