decaying states and the mathematics of quantum physics

REPORTS ON MATHEMATICAL PHYSICS

Vol. 67 (2011)

No. 2

RESONANCES/DECAYING STATES AND THE MATHEMATICS OF QUANTUM PHYSICS∗ A RNO B OHM Center for Complex Quantum Systems and Center for Particles and Fields, Department of Physics, University of Texas at Austin, Austin, Texas 78712-1081, USA (e-mail: [email protected]) (Received December 16, 2010) There is sufficient experimental evidence that a Breit–Wigner scattering resonance of width  is the same physical entity as an exponentially decaying Gamow state of lifetime τ = h/ ¯ . In order to derive a Gamow ket with exponential time evolution from the Breit–Wigner scattering amplitude of the S-matrix pole, one has to make assumptions about the mathematical properties of the energy wave function for the prepared in-state φ + and the detected out-“state” ψ − of a resonance scattering experiment. These mathematical properties identify the space of in-state 2 and of out-state wave functions as {ψ − (E)} = H2 as energy-wave functions as {φ + (E)} = H− + the Hardy function spaces of the lower and upper complex energy plane. The semigroup-time asymmetry t0 = 0 < t < ∞ of causality is a consequence of the Paley-Wiener theorem for Hardy spaces. The experimental meaning of the beginning of time t0 will be discussed. Keywords: Hardy space axiom, Paley Wiener semi-group, time asymmetric quantum theory, lifetime-width relation for quasi-stable particles, Gamow states, Breit–Wigner resonances.

1.

Introduction This paper is a sequel to a paper [1] delivered at the 40th symposium in Toru´n. The paper [1] presented the development of the mathematical theory of quantum physics starting with the matrices of Born, Heisenberg, Jordan, Pauli and with Schr¨odinger’s wave equation, culminating in von Neumann’s Hilbert space H. The Dirac bra-ket formalism inspired a new development in mathematics, Schwartz’s distribution theory and the Schwartz–Rigged–Hilbert space [Schwartz RHS] (also called Gelfand triplets) (1)  ⊂ H ⊂ × . This Schwartz space triplet gave a mathematical meaning to the Dirac ket |E as a continuous antilinear functional |E ∈ × . Quantum theory falls roughly into two categories: Category I problems are: The description of spectra and structure of microphysical systems. It applies to stable states and is also used for slowly decaying states when ∗ Presented

at the 42nd Symposium on Mathematical Physics, Toru´n, 19–22 June, 2010. [279]

280

A. BOHM

the finiteness of their lifetime is ignored. The energy values are discrete and real, and the time evolution is assumed to be the unitary group evolution. This is well described in the Schwartz RHS. In von Neumann’s Hilbert space as well as in the RHS (1), one does not distinguish mathematically between the set of in-state vectors {φ} defined experimentally by a preparation apparatus (prepared in-state φ + of a scattering experiment) and the set of observables {ψ} (detected out-states ψ − of a scattering experiment). Both are represented by the Schwartz space : {φ} = {ψ} = ,

{φ + } = {ψ − } = .

(2)

The observables denoted by A, B, ... or also |ψψ | are represented by an algebra of continuous operators in , which is not possible in the Hilbert space where the operators are usually unbounded. Thus one has an algebra of continuous observables in  and in × , but not in H. Category II problems of quantum mechanics are: Description of scattering, resonance phenomena and decaying states. This applies to states that are “rapidly decaying” and their time evolution is nontrivial; physical examples are doubly excited states of atoms (Auger states), nuclei, scattering resonances, and relativistic unstable particles. The Schwartz RHS (1) works for category I problems, it is not adequate for the quantum theory of scattering, resonances, and decay. Using the Hilbert space or the Schwartz space triplet, it is not possible to obtain a mathematical theory of resonance and decay phenomena because experimentally decaying states, Breit– Wigner resonances and similar objects are not possible in either of the three spaces of the Schwartz space triplet (1). The conventional mathematical theory for quantum physics is time-symmetric, the time t extends over −∞ < t < +∞. This is a consequence of the boundary conditions for the dynamical equations, the Heisenberg equation for observables |ψψ | or , ∂ ∂ ψ(t) = −H ψ(t), i h¯ (t) = −[H, (t)] ∂t ∂t as well as the Schr¨odinger equations for states φ i h¯

i h¯

∂ φ(t) = H φ(t). ∂t

(3)

(4)

To find the solutions of a differential equations, one needs to impose boundary conditions. The boundary conditions state, which kind of functions or which kind of vectors are admitted as solutions of the differential equations. In standard quantum mechanics, these boundary conditions are given by the Hilbert Space Axiom Set of states

{φ} = H = Hilbert space,

Set of observables

{ψ} = H = Hilbert space.

(5)

281

RESONANCES/DECAYING STATES

This means the wave functions E|φ = φ(E), E |ψ = ψ(E) are postulated to be Lebesgue square integrable. To one state φ does not correspond one wave function φ(E) but infinitely many that can differ from each other, on a set of measure zeros, e.g. at all rational numbers E 1 . From the Hilbert space boundary condition (5) follows (by the Stone–von Neumann theorem [2]) that the solutions of (4) are given by φ(t) = U † (t)φ = e−iH t/h¯ φ,

−∞ < t < ∞,

(6)

and the solutions of the Heisenberg equation (3) are given by |ψ(t) = U (t) |ψ = eiH t/h¯ |ψ,

with − ∞ < t < ∞.

(7)

For an operator  in (3), (t) = eiH t/h¯  e−iH t/h¯ ,

−∞ < t < ∞.

(7a)

This means that form axiom (5) follows that the time extends for states and for observables over −∞ < t < +∞. But one could choose to solve the dynamical equations (3), (4) using in place of (5) different boundary conditions. For instance, in the mathematical version of the Dirac formulation, one chooses the Schwartz space  for the set of state vectors {φ} as well as for the set of observable vectors {ψ}, i.e. one chooses (2) as the boundary conditions of (3) and of (4). 1. The solutions of both the Heisenberg equation (3) as well the Schr¨odinger equations (4) (state and observable) have the Dirac basis vector expansion,  ∞  ∞ dE |E, j, j3 , ηE, j, j3 , η |φ = dE |EE |φ, (8) φ= j,j3 ,η

0

0

where the basis vectors | E =| E, j, j3 , η are “eigenkets” or “generalized eigenvectors” of H (and the complete set of observables H , J 2 , J32 , η, which we often ignore) and these eigenkets fulfill φ | H |E, j, j3 , η = Eφ |E, j, j3 , η

for all vectors φ, ψ ∈ .

(9)

To each vector φ ∈  then corresponds one function E |φ = φ(E), not a class of functions as is the case for the Hilbert spaces axiom (5). 2. The functions, the co-ordinates of the vectors φ along the direction of the basis ket |E, E |φ = φ(E) are smooth, rapidly decreasing functions of E (“Schwartz function” ∈ S ). One has a triplet of function spaces {φ(E)} = S ⊂ L2 ⊂ S ×

(10)

and corresponding to it a triplet of vector spaces {φ} =  ⊂ H ⊂ × ,

(11)

1 This means, a smooth Schwartz function f (E) and another function, which differs from the smooth Schwartz function at all rational numbers, and is infinity at the rational numbers represent the same vector f ∈ H. Experimentally the probabilities | f (E) |2 E are obtained only for finite size intervals Ei .

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A. BOHM

called a Gelfand Triplet or Rigged Hilbert Space (RHS). The Dirac kets | E are antilinear continuous Schwartz space functionals | E ∈ × . The Dirac basis vector expansion (8) is the “nuclear spectral theorem” for the Schwartz-rigged Hilbert space, which can be defined (often called “realized”) by the Schwartz functions (10). In the mathematical version of the Dirac formalism, one uses the same RHS for the set of states {φ} and for the set of observables {|ψψ |}: {φ} = {ψ} =  = abstract Schwartz space.

(12)

The solutions of the dynamical equations, the Sch¨odinger equation (3) and the Heisenberg equation (4), are for the Schwartz space boundary condition, φ ∈ , ψ ∈ , also given by two groups similar to (6) and (7), as proven in [3]. From the boundary condition φ ∈  one obtains for the solution of (4) φ(t) = e−iH t/h¯ φ(0),

−∞ < t < +∞.

(13)

From the boundary condition ψ ∈  one obtains for the solution of (3) ψ(t) = eiH t/h¯ ψ(0),

−∞ < t < +∞.

(14)

For the Dirac kets one has then eiH

× t/h

¯

|E = eiEt/h¯ |E,

−∞ < t < +∞.

(15)

The conclusion is: For standard quantum theory, even when amended with the Dirac formalism using the Schwartz-rigged Hilbert space (12), one has a group evolution, (13) for the states, or (14) for the observables. This means: the quantum mechanical probability to detect, an observable (t) =|ψ(t)ψ(t) | in the state φ, which is given by the Born probability (Heisenberg picture),

Pφ ((t)) = Tr(|ψ(t)ψ(t) ||φφ) |= |ψ(t)|φ|2 = |e−iH t/h¯ ψ(0)|φ + |2 ,

(16)

is predicted for the time −∞ < t < +∞. But what is the experimental evidence for this time evolution group (13), (14); or (6), (7) in case the original Hilbert space axiom (5) in used? 2.

The quantum mechanical arrow of time

In quantum physics, the quantities measured in experiments are the probabilities for an observable represented by an operator  in a state represented by an operator W (a special case is  =|ψψ | and W =|φφ |, then φ is called the state vector and ψ the observable vector). The quantities that are measured are the Born probabilities Tr( W ) which are measured as ratios of large integers Ni /N of detector counts Ni ( Ni = N), see Eq. (68) of [1]. For instance, the probability to register with a detector in a scattering experiment the observable |ψψ | in the prepared in-state φ is given by the Born probability

Pφ (|ψψ |) = Tr(|ψψ |φφ |) = |ψ |φ|2 .

(17a)

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RESONANCES/DECAYING STATES

In scattering experiments one has an in-state φ + prepared by a preparation apparatus (accelerator, etc.) and a detected observable |ψ − ψ − | registered by e.g. a counter; the counts N (ψ − )/N are predicted to be proportional to

Pφ + (|ψ − (t)ψ − (t) |) = |ψ − (t)|φ + |2 = |ψ − |φ + (t)|2

for −∞ < t < +∞.

(17)

In general, according to one of the axioms of quantum theory, the Born probabilities are calculated as

PW ((t)) = Tr(W (t) ) = Tr(W (t)) ≈ N (t)/N,

(18)

and measured as the counting rates N (t)/N of detectors. The first expression on the r.h.s of (18) is in the Heisenberg picture and the second is in the Sch¨odinger picture. The comparison (18) of the calculated Born probability P ((t)) and the experimental detector counts N (t)/N applies to ensembles of N detector counts, where N and N (t) are “large” numbers (in some high energy experiments one has 106 –109 events, in others only about 10 to 100). What is the experimental evidence for the time evolution group (13), (14)? It is obvious that a state φ must be prepared before the observable (t) = |ψ(t)ψ(t) | can be measured in it. This is causality. The detector cannot count decay products of a decaying state before the decaying state has been prepared. Thus, we have a quantum mechanical arrow of time: The Born probability to measure the observable |ψψ | in the state φ

Pφ (ψ(t)) = |ψ(t) |φ|2 = |eiH t/h¯ ψ |φ|2 = |ψ |e−iH t/h¯ |φ|2 = |ψ |φ(t)|2

(19)

exists (experimentally) only for t ≥ t0 (= 0). Here t0 denotes a time at which the state φ has been prepared and at which the detector can register the observables |ψψ | in the state φ. The principle subject of this paper is to understand the mathematical definition and the experimental observation of this “quantum mechanical beginning of time” t0 . This statement (19) is in contrast to the mathematical prediction in (17) which followed from (13) or(14) for the Schwartz space axiom (2), and from (6), (7) for the Hilbert space axiom (5). Thus the Hilbert space axioms (5) as well as the Schwartz space axioms (12) are in conflict with some intuitive principle of causality (19) which states: A state must be prepared first by t0 before an observable can be detected (registered) in the state for t ≥ t0 .

(20)

The unitary time evolution (6), (7) of the Stone–von Neumann theorem has been associated with reversibility on the micro-physical (atomic, nuclear, relativistic particle) level; it has been our understanding of the quantum world. Therefore the causality statement (20), expressed by the quantum mechanical arrow of time (19), was usually not considered applicable to quantum dynamics on the microphysical level.

284

A. BOHM

Despite the preponderance of the unitary group representation (6) and its reversible time evolution, many people were aware of some kind of quantum mechanical time asymmetry [4–8]; e.g.: * Feynman [5] says “We choose a particular time t = t0 and divide the region . . . into (a) a region R such that t < t0 and . . . (b) a region R

such that t

> t0 .” “The state at t is defined completely by the preparation (prepared in-state φ + ). . . Likewise the state characteristic of the experiment (region R

) (registered observable ψ − or out-state) can be defined by ψ(t

),. . . at time t

> t0 .” “the chance that the system prepared in state φ(t ) at time t will be found after t

to be in a state (characteristic of experiment) ψ(t

) is the square of the transition amplitude” (ψ(t

), φ). (The symbols are chosen to agree with those used in this paper, the words are Feynman’s). He clearly distinguishes between prepared states and “states characteristic of the experiment” in place of the latter we use the word

observables. And he chooses a time t0 after which, for t > t0 , the transition

probability |(ψ(t ), φ)|2 will be found. * T. D. Lee [6] (1981) called this time asymmetry the “impossibility of constructing time-reversed quantum solutions for the micro-physical system” (in muondecay). * Gell-Mann and Hartle [7] (1990) introduced an arrow of time in the quantum theory of cosmology. In order to avoid inconsistencies for the probabilities of histories, they required a time ordering for the projection operators in a history and the initial states. This time ordering “may not be attributed to the thermodynamic arrow of an external measuring apparatus or larger universe”, but it “would be a fundamental quantum mechanical distinction between the past and the future”. The “experimental evidence” for this time asymmetry is the Big Bang, at which time our Universe was a quantum system. * Peierls [8] and his school declared that time asymmetry does not arise from the dynamical equations of motion (e.g. the Schr¨odinger or Heisenberg equations of motion) but from the boundary conditions for these equations, and that the situations which arise in nature are usually described by initial conditions, not by terminal conditions. Peierls’ reason is that one performs experiments in which initial conditions are specified. The best known example of time asymmetry from classical physics is the radiation arrow of time: Maxwell’s equations (dynamical differential equations) are symmetric in time. The Sommerfeld radiation condition, a boundary condition, excludes the strictly incoming fields μ (21a) Ain (x) = 0, selects only the retarded fields of the other sources in the region μ

μ

μ

Aμ (x) = Aret (x) + Ain (x) = Aret (x).

RESONANCES/DECAYING STATES

285

The boundary condition is an additional “principle of nature” that chooses of the two solutions of the Maxwell equations,  μ

  j (x , t ) 3

μ



d x dt , x , t) = δ t − (t ∓ | x − x |/c) A∓ ( | x − x | the retarded solution



μ x , t) Aret (

≡

μ A− ( x , t)

=

j μ (x , t − | x − x |/c) 3

d x. | x − x |

(21b)

The “disturbance” Aμ (x) at the position x at time t is caused by the source j μ at another point x , at an earlier time x − x |/c ≤ t. t = t − | A source (transmitter) must emit radiation (at t ) before the radiation can be detected by a receiver at t ≥ t (causality). The quantum mechanical arrow of time (19) is the quantum mechanical analog of this radiation arrow of time. All these considerations suggest that one needs to find a theory in which the Hilbert space boundary conditions (5) are replaced by new boundary conditions which in the Heisenberg picture lead to time translated observables ψ − (t) = eiH (t−t0 )/h¯ ψ −

defined only for t > t0 ,

(22)

and which in the Schr¨odinger picture leads to time evolved states φ + (t) = e−iH

× (t−t )/h + 0 ¯φ

defined only for t > t0 .

(23)

Here t0 is the preparation time of the state φ + (t0 ) = φ + ; it is a finite value for which one often chooses t = 0. This means their time evolutions are asymmetric, t0 ≤ t < ∞, and given by the semigroup:

U × (t) = e−iH

× (t−t )/h 0 ¯

with t0 ≤ t < ∞ for the states φ or ρ

(24)

or by the semigroup

U (t) = eiH (t−t0 )/h¯

with t0 ≤ t < ∞ for the observables ψ or .

(25)

Thus, the task is, to search for a theory with the same dynamical equations (3) and (4) but with new boundary conditions, other than the Hilbert space axiom (5) or the Schwartz space boundary condition (12). These new boundary conditions will have to be chosen such that the solutions of the Schr¨odinger equation, φ(t), are given by the semigroup U × (t) of (24), and the solutions of the Heisenberg equation, ψ(t), are given by the semigroup U (t) of (25).

286

A. BOHM

3.

New boundary conditions for the dynamics equation In order to obtain these two different time evolutions (23) of the states and (22) of the observables, we need to find new boundary conditions for the Schr¨odinger equation (4), or for the Heisenberg equation (3). This means to find a new axiom, which will replace the Hilbert space axiom (5) and the Schwartz space axiom (12). This new axiom needs to say, that the set of prepared states {φ + } is given by a new space which we denote − , and the space of detected observables {ψ − } is given by another new space + such that the time evolution is given by the semigroup evolution (22) for the ψ − ∈ + and by (23) for φ + ∈ − : The set of prepared (in-) states defined by the preparation apparatus (e.g. accelerator) is {φ + } = − ⊂ H ⊂ × −. The set of (out-) observables defined by the registration apparatus (e.g. detector) is {ψ − } = + ⊂ H ⊂ × +.

(26) (27)

Both spaces − and + are chosen to be dense subspaces of the (same) Hilbert × space H, and need to have different (dual) spaces {× − } and {+ }, respectively. The triples (26), (27) are rigged Hilbert spaces or Gelfand triples.2 The notation φ + ∈ − and φ − ∈ + may appear curious, it has its origin in two different conventions: the spaces ± have their origin in mathematics. We shall show below that ± are Hardy-spaces of the upper and lower complex plane, respectively [9, 10]. The labels ∓ at the vectors and the kets are taken from the (most common) convention of scattering theory in quantum mechanics. (Except that the ψ − are considered here to be observables fulfilling the Heisenberg equation (3) rather than called out-states.) The “nuclear spectral theorem” for the rigged Hilbert spaces (26), (27) provides the eigenkets |E ±  for the Dirac basis vector expansions of in-state vectors  ∞  ∞ + + + + dE|E + + E|φ + , (28) dE|E, j, j3 , η  E, j, j3 , η|φ  = φ = j,j3 ,η

0

0

and for the Dirac basis vector expansion of the out-vectors for the observables |ψ − ψ − |,  ∞  ∞ − − − − ψ = dE|E, j, j3 , η  E, j, j3 , η|ψ  = dE|E − − E|ψ − , (29) j,j3 ,η

0

0

here j, j3 , η denote the additional quantum numbers: j, j3 are usually angular momentum quantum numbers, and η are particle species labels, e.g. channel quantum numbers. The existence of the kets |E ±  =|j, j3 , η±  follows from our hypothesis that the spaces ± are, like the Schwartz space (11), spaces in which the nuclear spectral theorem holds. The use of two kinds of kets |E ±  was already suggested by the Lippmann–Schwinger equations [11], cf. Eq. (70) of [1]. The task is then to find 2 Then

the nuclear spectral theorem will hold, as in (8) for the Schwartz triplet (11).

RESONANCES/DECAYING STATES

287

these two RHS’s (26) and (27), this means to provide the mathematical definition of these two triplets (26) and (27). In the same way as the Schwartz space {φ} can be defined by the space of Schwartz functions {φ(E) = E | φ}, the pair of spaces (26) (27): {φ + } = − and {ψ − } = + are defined by the spaces of functions {φ + (E) = + E |φ + } and {ψ − (E) = − E |ψ − }, their “components or coordinates”.3 To find the RHS’s (26) and (27) is thus equivalent to finding the function spaces {φ + (E)} and {ψ − (E)}. In order to determine (“conjecture”) the properties of these functions, one has to use physical results. The Dirac basis vector expansions (28) and (29) use two different kinds of kets: |E ±  = |Ejj3 η∓  ∈ × ±,

(30)

as had been already suggested by the Lippmann–Schwinger out-plane waves |E −  and in-plane waves |E + , Eq. (70) of [1]. Because of the +i in the Lippmann–Schwinger equation, the energy wave function of the in-state φ + prepared by the preparation apparatus (e.g. accelerator), φ + (E) ≡ + E |φ +  = + Ejj3 η|φ +  = φ + |Ejj3 η+ 

(31)

is the boundary value of an analytic function in the lower complex energy semi-plane (for complex energy z = E + i = E − i immediately below the real axis). One uses the second sheet of the S-matrix Sj (E), cf. Section 5.4 of [1]. Similarly, the energy wave function of the observable |ψ − ψ − |, ψ − (E) ≡ − E |ψ −  = − Ejj3 η|ψ −  = ψ − |Ejj3 η− 

(32)

can be extended into an analytic function in the upper complex energy plane, than its complex conjugate ψ − |E −  is analytic in the lower complex plane second sheet of the S-matrix. Thus we have spanning the from two sets two sets of for the two two spaces of L-Sch. kets functions sets of vectors (33) + + + + + + − = {φ + } |E  = |Ejj3 η  {φ (E) =  E |φ } φ = states + = {ψ − } {ψ − (E) = − E |ψ − } ψ − = observables |E −  = |Ejj3 η−  The mathematical properties of the two sets of functions {φ + (E)}, {ψ − (E)} are conjectured from the physical properties that they are expected to describe. The φ + are interpreted as prepared in-states of a scattering experiment and as states they fulfill the Schr¨odinger equation (4). The ψ − are in standard scattering theory interpreted as out-states of the scattering experiments, and as states they should fulfill the Schr¨odinger equation (4). However, this is a mis- interpretation. The operator  =|ψ − ψ − | in the Born probability Tr( W ) = |ψ − |φ + |2 represents the detector which registers the observable. The operator  =|ψ − ψ − | in the Born 3 Like

 the components x i = (e i · r ) of the vector r = 3i=1 e i · x i in 3-dimensional space.

288

A. BOHM

probability (18), since it represents the observable, needs to fulfill the Heisenberg equation (3). The operator W =|φ + φ + | represents the prepared state and fulfills the von Neumann equation, and the state vector φ + fulfills the Schr¨odinger equation (4). The state is defined by the preparation apparatus (e.g. accelerator) of a scattering experiment. The observable |ψ − ψ − |, or the “observable-vector” ψ − (t), represents the detector of a scattering experiment and therefore needs to fulfill the Heisenberg equation (3). The time dependence of the Born probabilities is then given by (19) as it needs to be; the first line of (19) is in the Heisenberg picture and the second line of (19) is in the Schr¨odinger picture. This justifies the modification in our interpretation of the ψ − from “out-state” to “out-observable”.4 In the same way as the Schwartz space  = {φ} of the Dirac formalism is mathematically defined by the space of the Schwartz functions {φ(E) = E |φ}, the space − is mathematically defined by the space of functions {+ E |φ + }

(33a)

and the space + is mathematically defined by the space of functions {− E |ψ − }.

(33b)

We want to find the new boundary conditions for the dynamical equations (3) and (4), i.e. find the spaces − and + , and therewith the triplets (26) and (27) for states and observables. This can only be done phenomenologically by comparing the results of theoretical considerations with prominent experimental results. To conjecture this theory we use resonance and decay phenomena. 4. Determining the spaces of states and observables The question is: What are the two spaces {φ + } = − , {ψ − } = + ? Or, equivalently, what precisely are the spaces of their energy wave functions {ψ − (E) = − E |ψ − } and {φ + (E) = + E |φ + }? The answer of this question can be obtained from physical observations and experimental results. The vectors ψ − and φ + appear in the probability amplitude ψ − |φ +  of the observable |ψ − ψ − | in a state φ + . The state is prepared by a preparation apparatus (e.g. an accelerator), the observable is registered by the detector. This probability amplitude is described by the S-matrix, Eqs. (78), (79) of [1], ∞ out in − + (34) (ψ , S φ ) = (ψ | φ ) = dE ψ − |E − Sj (E)+ E |φ + . E0

Here Sj (E) is the j -th partial S-matrix element which encapsulates the dynamics of the scattering process. 4 Since in cross-section measurements the observable ψ − is always registered at its time t = 0 (which is an ensemble of times by the clock in the lab), the mis- identification of ψ − (t) as out -“state” rather than an out-observable did not lead to catastrophic consequences.

RESONANCES/DECAYING STATES

289

We now consider the case that the S-matrix element Sj (E) has one pole at zR = ER − i/2. Then the contour of integration can be deformed from the positive real axis first sheet = positive real axis second sheet into the lower complex energy plane second sheet, where the resonance pole of the S-matrix is located, at zR = ER − i/2, cf. Fig. 1 of [1]. The Lippmann-Schwinger equation [11] suggested by the ± i in (70) of [1] that the − E |ψ − , + E |φ +  are boundary values of analytic function. From S-matrix theory [12] one knows that the energy in (34) has to be continued into the complex plane second sheet of the S-matrix.5 In many cases of scattering experiments, one observed that the cross section, e.g. the partial cross section of angular momentum j , σj (E), has a bump at a particular value of the energy E = ER . This bump has the characteristic features of a Lorentzian or Breit–Wigner as shown in [1]. This Breit–Wigner line-sharp is an indication that: 1. The energy E in the integrand in (34) needs to be analytically continued to complex values, specifically to complex values on the second sheet of the Riemann surface for the S-matrix element Sj (E).

ηη

2. The partial wave amplitude, aj (E) ∼ Sj (E) (for the channel quantum numbers

η = η) is given by the Breit–Wigner r , zR = ER − i/2. (35) ajR (E) = E − zR Since the Breit–Wigner with the same values of (ER , ) appear in several different channels (for different values of the particle species or channel quantum numbers

η , η); this is an indication that there should be states characterized by a complex energy value zR , and it introduces a new parameter with the dimension of energy, the width  of the Lorentzian (35). From (35), one concludes that the energy wave function (31) in (34) of the prepared in-state φ + , is an analytic function in the lower complex energy semi-plane. Similarly, one concludes that the energy wave function (32) in (34) of the observable can be extended to an analytic function in the upper complex energy semi-plane. (So that the integrand in (34) can be analytically continued into the lower complex plane second sheet of Sj (E).) To determine the space − of prepared states and the space + of registered observables is—because of (28), (29)—equivalent to determining the space of their energy wave function φ + (E) = + E |φ +  and ψ − (E) = − E |ψ − . To determine the two function spaces {+ E | φ + } and {− E | ψ − } we use physical properties of decay and resonance phenomena. Specially, we shall use the hypothesis that scattering resonances defined by (35) are the same physical entities as exponentially decaying states. 5 See

Fig. 1 of [1].

290

A. BOHM

Resonances are measured by the Breit-Wigner lineshape in the cross section  2  r  2 σj (E) ∼ |aj (E)| =  zR = ER − i/2, (36) + B(E) , E − zR where B(E) is a slowly varying function of E describing the non-resonant background in the scattering process, or the influence of other far away resonances. The resonance (35) is characterized by energy and width (ER , ). It is associated with the first order pole of the S-matrix, Fig. 1 of [1]. Decaying states are defined by the counting rate of the decay products η, the time derivatives of the probabilities (19), d P (ψη (t)) . (37a) dt The decaying state is characterized by energy ED and exponential lifetime τ , which is measured by a fit of the counting rate to the exponential function e−t/τ , Rη (t) =

Rη (t) ≈

N (ti ) ∝ e−t/τ .

ti

(37b)

Here N (ti ) is the number of decay products registered in the detector during the time interval ti around ti . The signature of a decaying state is the exponential in (37b); the signature of a resonance is the Breit–Wigner (35) in (36). Many physicists think that resonances (ER , ) are the same as decaying states (ED , τ ). Especially for nonrelativistic quantum mechanics, a common assumption is that h¯ ≈ τ.  This opinion is based on the Weisskopf–Wigner (WW) approximation [13]. But in standard quantum theory there is no proof of h¯ /  = τ .6 Using WW methods, the probability to observe the decay products |ψψ | in a prepared resonance state φ(t) with Breit–Wigner lineshap of width  is calculated in [14] as |ψ |φ(t)|2 = Pφ(t) (ψ) ∼ e−t +  × (additional terms).

(38)

In the Hilbert space (5) there is no vector with exactly exponential time evolution, i.e. no vector φ G for which the (additional terms) in (38) can be zero and which would predict the exponential law (37b) [15]. Also in the dual of the Schwartz space, the space × of (11), there does not exist a vector φ G ∈ × , for which Pφ G (t) = |ψ |φ G (t)|2 = |ψ(t) |φ G |2 obeys the 6 M. Levy: “... There does not exit... a rigorous theory to which these [WW] methods can be considered as approximation”, Nuovo Cimento 13 (1959), 115.

RESONANCES/DECAYING STATES

291

exponential law. Such a vector would lead to the “exponential catastrophe” [17] due to the fact that in H as well as in × the time extends over −∞ < t < +∞. To obtain a vector φ G with perfect exponential time evolution, we need a Gamow vector φ G , i.e. a vector with the property H φ G = (ER − i/2)φ G

and φ G (t) = e−iH t/h¯ φ G .

(39)

Then its decay probability (17) into any observable |ψψ | would be given by the exponential law:

Pφ G (t) (ψ) = |ψ |φ G (t)|2 = |ψ | e−iH t/h¯ |φ G |2 = |ψ |φ G e−i(ER −i/2)t/h¯ |2 = |ψ |φ G |2 e−t/h¯ .

(40)

One can prove that in Hilbert space, there is no vector φ(t) that obeys the exact exponential law [15]. Such vectors also do not exist in × (or ). But such vectors × could exist in × − or in + , the duals of the new RHS’s (26) or (27). But, if such × vectors do exist in − or × + , we would expect, on the basis of the above remarks concerning the exponential catastrophe, that in these spaces the time evolution could not be given by a group with t extending over −∞ < t < +∞. In contrast to the historical prediction of the W–W methods (38) and the other theoretical argument, recent experiments show with ever increasing accuracy that spontaneously decaying quantum systems obey the exponential law P (t) ∼ e−t/τ . For the standard review see [19]. In particular there are: 1) The KS0 -decay experiments, e.g. [18] (KL0 interference analyzed out) show that the time dependence of the decay rate (37b) is perfectly exponential. This 0 means the probability: |ψπ π |φ Ks (t)|2 = N (ti )/N is a perfect exponential, e−t/τ . 2) The relation τ = h/ ¯  has been tested recently for the 32 P3/2 level of Na to an accuracy that goes beyond the Weisskopf–Wigner approximation. Both linewidth  [20] and lifetime τ [21] have been measured with an unprecedented accuracy, with the result: linewidth measurement gives h/ ¯  = (16.237 ± 0.035) ns, lifetime measurement gives τ = (16.254 ± 0.022) ns.

(41)

Summarizing the empirical situation: There is sufficient experimental support for the hypothesis that a resonance represented by the Breit–Wigner (35) (without the background term B(E) in (36)) and the exponentially decaying state represented by a Gamow ket with the property (39), (40) (but without the (additional term) in (38)) represent the same physical entity, Breit–Wigner resonance ≡ exponentially decaying Gamow ket.

(42)

We shall therefore associate to the resonance, the Breit–Wigner amplitude (35) which is given by the pole term of the S-matrix element Sj (E) in (79) of [1]. For

292

A. BOHM

simplicity, we consider the case that the S-matrix element Sj (E) has just one pole as shown in Fig. 1 of [1]. The integral in (79) along the upper edge of the cut can be deformed through the cut into the second sheet with the result that one obtains an integral along C− , and the integral along the circle C1 around the resonance pole zR and the integral along the infinite semicircle C∞ . The spaces of wave functions {+ E |φ + } and {− ψ |E − }, which we want to find, for sure must be such that the integral along C∞ vanishes. Then there is the integral along C− , it represents nonresonant background scattering which has nothing to do with the resonance at zR . We shall work with the integral around the resonance pole zR in order to determine the mathematical spaces of the energy wave functions − = {+ E |φ + }

and

+ = {− ψ |E − }.

(43)

The physical principle that we want to use for the determination of these mathematical spaces is that: a Breit-Wigner resonance (35) with (ER , ) is the same physical entity as an exponential decaying Gamow state with (ED , τ = h/ ¯ ), as exemplified by the experimental data in (41). It will turn out that these mathematical spaces derive from well-known spaces in functional analysis of the previous century [9], and the Gamow state vector is a ket, i.e. an antilinear continuous functional on one of these spaces. 5.

Conjecturing the Hardy spaces for quantum scattering resonance and decay theory We start with the definition of a resonance as the Breit–Wigner (35) at the pole position zR of Sj (E) in (34). Then we deform the contour of integration (as shown in Fig.1 of [1]) through the cut along positive real axis = spectrum of H in (34), into an integral around the pole at zR = ER − i/2 on the second sheet and an integral along the contour C− . (The integral along C∞ will be vanishing and is of no interest to us at the moment). The integral (34) then becomes   s−1 + − + − − + + dωψ |ω S(ω) ω |φ  + dωψ − |ω−   ω |φ +  (44) (ψ , φ ) = ω − zR C−  where s−1 = i is the residue7 of S(ω) at zR , i.e. s−1 + S0 + S1 (ω − zR ) + · · · . (45) SI I (ω) = ω − zR The arrow at the second integral indicates that the direction of integration around zR is as shown in Fig.1 of [1]. We are interested in the second integral of (44) since it is associated with the resonance pole, it can be evaluated using the Cauchy  theorem, 1 f (ω) , f (zR ) = dω 2π i ω − zR 7 That

s−1 = i follows from the unitarity of the S-matrix with one pole.

293

RESONANCES/DECAYING STATES

to give the pole term ψ

−

|zR− (2π )+ zR |φ + 

 =



dωψ − |ω− + ω |φ + 

i . ω − zR

(46)

The first integral on the right-hand side of (44) represents some background integral not related to the resonance, which is denoted by B(E) in (36). If the energy wave function ψ − | ω−  and + ω | φ +  fulfill some conditions in addition to being “well behaved” functions, (i.e. − E |ψ − , + E |φ +  ∈ S, or ψ − , φ + being ∈ ) the contour of integration in (44) and (46) can be deformed further. Because of our condition that the pole of the S-matrix be associated to a Breit–Wigner resonance amplitude (35), we want to require that the pole term of the S-matrix element in (44), i.e. the integral in (46), is given by a Breit-Wigner integral. This means the analytic function ψ − |ω− + ω |φ +  in (44) needs to vanish sufficiently fast on the infinite semicircle second sheet, so that the right-hand side of (46) becomes the right-hand side of (47) below:  +∞ i − − + + dEψ − |E − + E |φ +  . (47) ψ |zR  zR |φ 2π  = E − (ER − i/2) −∞I I In order that the right-hand side of (46) leads to the right-hand side of (47), the energy wave function must fulfill certain mathematical conditions which identify them as Hardy functions. Precisely, in order that (47) follows from (46) the energy wave functions must fulfill: 2 ∩S − E |ψ −  ∈ H+ +

+

 E |φ  ∈

2 H−

∩S

or or

2 ψ − |E −  ∈ H− ∩ S, +

+

φ |E  ∈

2 H+

(48)

∩ S.

(49)

2 2 (Hardy class from above) and H− (Hardy class from below) denote two Here, H+ spaces of functions which are boundary values of analytic functions on the upper and lower half planes respectively, cf. [3, 9, 10], and S is the Schwartz function space. To summarize: if (48) and (49) are fulfilled, then the contour of integration on the right-hand side of (46) can be deformed into the contour of integration in (47) from −∞ to +∞ along the (lower edge in the second sheet of the ) real axis, cf. Fig 1 of [1]. If the wave functions have the respective properties (48) and (49), then we call the ψ − of (29) “very well-behaved vector from above” and we call the φ + of (28) “very well-behaved vector from below.” This we write as8

ψ − ∈ +

(50a)

φ + ∈ −

(51a)

8 Note that the − and + at the vectors has its origin in the standard notation of scattering theory and the 2 + and − at the spaces has its origin in the standard notation in mathematics for the Hardy class spaces H+ 2 and H− .

294

A. BOHM

Only positive values of E correspond to physical values of the (scattering) energy. Thus, the boundary values of ψ − |ω−  and + ω |φ +  are physically determined only on R+ = the upper rim on the cut of the Riemann surface in Fig. 1 of [1]. But from mathematical theorems, the Titchmarsh theorem and the van Winter theorem, these physically determined boundary values on R+ completely determine the Hardy class functions (and therewith also the smooth Hardy functions (48) (49)) on the whole complex semi-plane [3, 9, 10]. We therefore can restrict ourselves to the spaces of smooth Hardy functions on the positive real axis (the energy spectrum 0 ≤ E ≤ ∞ along the cut in Fig. 1 of [1]). Therewith we have arrived at the following definition for the spaces of energy wave functions in (28), (29), 2 ∩ S |R + , − E |ψ −  = ψ − (E) ∈ H+

(50b)

2 ∩ S |R + + E |φ +  = φ + (E) ∈ H−

(51b)

−

+

Since these smooth Hardy functions {ψ (E)} and {φ (E)} are subspaces of the Schwartz space S, which is a nuclear space, the spaces (50a) and (51a) are also nuclear spaces. Therefore, the Dirac basis vector expansions (28) and (29), with the energy wave functions (50b), (51b) are justified by the nuclear spectral theorem for the Hardy spaces (48), (49). The nuclear spectral theorem (28), (29) with the energy wave functions given by (50b) (51b) is the mathematical justification to extend the Dirac formalism to Lippmann–Schwinger kets, Breit–Wigner resonances and exponentially decaying Gamow states. The mathematical Gamow ket |zR−  is defined as the Hardy space functional associated to the pole at zR = ER −i/2. Applying the Titchmarsh theorem [3, 9, 10] for Hardy functions to ψ − |E −  one obtains9  +∞ 1 1 − − 2 ψ |zR  = − dEψ − |E −  for ψ − |E −  = − E |ψ −  ∈ H− ∩ S. 2π i −∞I I E − zR (52) The integral extends along the negative real axis of the second sheet and then along the cut (spectrum of H ). Omitting the arbitrary ψ − ∈ + in (52), we obtain the equality between the functionals |E − , −∞ < E < +∞ and the Gamow ket |zR−  =|ER − i/2− ,    √ 1  1/2 +∞ 1 − dE |E −  |ER − i/2 ( 2π ) = − ≡ ψ G . (53) i 2π E − (ER − i/2) −∞I I This is the definition of the Gamow ket | zR−  ∈ × + . With the new limits of integration extending from −∞ to +∞, Eqs. (47) as well as (52) and (53) are 9 Here ∞ I I indicates that the contour of integration is on the second sheet from −∞ < E < 0 cf. Fig 1 of [1] and then along the cut over 0 ≤ E < ∞. Eq. (52) is the Titchmarsh theorem for the smooth Hardy function 2 ∩ S which is a special case of the Titchmarsh theorem for Hardy classes f (E) ∈ H2 . − E |ψ −  ∈ H+ +

RESONANCES/DECAYING STATES

295

consequences of the Titchmarsh theorem for Hardy functions10 (or Hardy kets in the case of (53)).11 This specific feature of the Hardy functions to extend the domain of integration to the whole real line −∞ < E < +∞, will allow us to show that the Gamow vectors (53) represent exponentially decaying states. Because of the ability to identify Hardy functions of the semi-planes with their boundary values on R+ (a consequence of the van Winter’s theorem [3, +∞9, 10] for Hardy functions), we can define the Gamow vectors by an integral −∞ dE, as II done in (52) and (53). The “exact” Gamow vectors (53) replace the “approximate” Gamow vectors of the conventional formulation, which are often defined by an +∞ integral as in (53), but with the integral extending over 0 dE, as prescribed by the spectrum of H . These approximate Gamow vectors, with the integration in (53) restricted to real energy 0 ≤ E ≤ ∞, do not exhibit perfect exponential decay. However, for /2ER  1 these approximate Gamow vectors resemble the exact Gamow vectors (52) and (53) arbitrarily closely. The Gamow kets (53) are uniquely determined by the poles of the analytically continued S-matrix. But, due to the “background” integral given by the first term on the right hand side of (44), the time evolution of a state vector φ + ∈ − always contains deviations from the exponential behavior; these have to be removed in the analysis from the experimental data, cf. e.g [18]. Therewith we have found the mathematical spaces (26), (27): they are defined as the two spaces of vectors: {φ + } whose continuous components {+ E |φ + } are the smooth Hardy functions (51b) of the lower complex plane12 and {ψ − } whose continuous components {− E |ψ − } are the smooth Hardy functions 2 ∩ S. (50b) of the upper complex plane, and − ψ |E −  = − E |ψ −  ∈ H+ − The Gamow kets | zR  defined by the Titchmarsh theorem (52) or (53) are eigenvectors of the Hamiltonian H and of H p for every p = 1, 2, 3, . . . To show this, we calculate H p |zR− ; we replace ψ − → H p ψ − in (52), and use H p ψ − |E −  = ψ − | H ×p |E −  = E p ψ − |E − . Then we obtain13 H ψ p

−

|zR− 

1 =− 2π i



+∞ −∞

dE E p ψ − |E − 

1 p = zR ψ − |zR− . E − zR

10 Here −∞ I I indicates the second sheet −∞I I < E < 0 in Fig. 1 11 It is the combination of the nuclear spectral theorem for S and the 2 that allows the representation (52) (53) of a Gamow ket |z− . H− R 12 Second sheet of the S-matrix.

(54)

of [1]. Titchmarsh theorem for the Hardy class

13 The

second equality in (54) is a consequence of the Titchmarsh theorem for Hp -spaces, i.e. for function p like G− (E) ≡ E p ψ − |E −  ∈ H− ∩ S [3, 9] since one has E p ψ − |E −  ∈ H− ∩ S p

if

2 ψ − |E −  ∈ H− ∩ S.

296

A. BOHM

Eq. (54) means that |zR−  ∈ × + is a generalized eigenvector of H with eigenvalue zR , in particular for p = 1 (and similarly for any integer p = 2, 3, . . . ), H ψ − |zR−  = ψ − | H × |zR−  = zR ψ − |zR− 

for all ψ − ∈ + . 14

(55a)

Written as a functional equation by omitting the arbitrary ψ − ∈ + —like for Dirac kets—this becomes: |zR−  ∈ × (55b) H × |zR−  = zR |zR− , +. As in case of the Dirac kets, which are functionals on Schwartz space H × |E = E |E, |E ∈ , one usually omits the × at H × and writes (55a) also as H |ER − i/2, j, j3 , η−  = (ER − i/2) |ER − i/2, j, j3 , η− , for 6.

|ER − i/2, j, j3 , η−  ∈ × + (55c)

Semigroup time evolution and time asymmetry

We have started from the hypothesis that scattering resonances identified by the Breit–Wigner (35) in the cross-section (36), and decaying states identified by the exponential in the decay rate (37b), are the same quantum physical entities. This is not possible using the mathematical tools provided by the standard mathematical theory of quantum physics based on the Hilbert space axiom (5) or in the mathematical theory of the Dirac formalism given by the Schwartz space axiom (12). In order to relate a Breit–Wigner resonance amplitude (35) to the ket |zR−  of (53), which is a continuous antilinear functional (similar to the mathematical Dirac ket), one has to distinguish mathematically between the set of state vectors defined by a preparation apparatus (in-state of a scattering experiment φ + ), and the set of observables (detected “out-state” of a scattering experiment ψ − but with the ψ − (t) obeying the Heisenberg equation (3) not the Schr¨odinger equation (4)). In order to obtain the relation (52), (53), which connect the Gamow ket and the Breit–Wigner resonance amplitude (35) one has to use the Titchmarsh theorem (52) for Hardy functions. This leads to the Hardy space axiom (50a,b) for observables and (51a,b) for states. The new axiom provides now the boundary condition for the dynamical equations (3) and (4). In the same way as one invokes for the Hilbert space axiom (5) the Stone–von Neumann theorem to obtain the unitary group evolution (6) and (7), one can in the case of the Hardy space axiom (50), (51) invoke another mathematical theorem, the 14 H ×

denotes the conjugate operator of H defined by H ψ − |F −  = ψ − | H × |F − 

† for all ψ − ∈ + and all F − ∈ × + ; it is an extension of the adjoint Hilbert space operator H . For self-adjoint Hamiltonian one has H ⊂ H = H† ⊂ H×

where H is the closure of the essentially self-adjoint H in the completion of + to H for the rigged Hilbert space + ⊂ H ⊂ × +.

297

RESONANCES/DECAYING STATES

Paley–Wiener theorem [23]. This mathematical theorem states that the solutions of the Heisenberg equation (3) for the Hardy space boundary condition are given by the semi-group ψ − (t) = eiH t/h¯ ψ − (0) with 0 ≤ t < ∞

for ψ − ∈ +

(Hardy above).

(56)

And the solution of the Sch¨odinger equation (4) are given by the semigroup φ + (t) = e−iH t/h¯ φ + (0) with 0 ≤ t < ∞

for φ + ∈ −

(Hardy below).

(57)

These semigroup evolutions bring a new concept into quantum theory, the beginning of time t0 = 0. This t0 is the same t0 = 0, that came up in Section 2 on the basis of causality (19) and it is probably related to the t of the radiation arrow of time (Section 2). It is remarkable that this causality has been more readily accepted in the relativistic theory as Einstein causality of the Poincar´e-semigroup transformations into the forward light cone

P+ = {(, x) : det = +1, 00 ≥ 1 ; x 2 = t 2 − x 2 = t 2 − r 2 /c2 , t ≥ 0}, than in nonrelativistic quantum mechanics, where the group evolution −∞ < t < +∞ is just a consequence of the mathematical Stone–von Neumann theorem for the Hilbert space. As a special example of the Paley–Wiener theorem, we shall consider the time evolution of the Gamow kets: The time evolution in Hilbert space H is represented by e−iH t/h¯ for the solutions of the Schr¨odinger equation or by eiH t/h¯ for the solutions ψ − (t) of the Heisenberg equation (22). The extensions of this operator to the two × different spaces × + and − are given by the definition of the conjugate operator; − we have for all ψ ∈ + , eiH t/h¯ ψ − |zR−  = ψ − | (eiH t/h¯ )× |zR−  = ψ − | e−iH

× t/h

¯

|zR− .

(58)

In order to define the conjugate operator (eiH t/h¯ )× of eiH t/h¯ in (58), the operator eiH t/h¯ needs to be a continuous operator in the Hardy space + = {ψ − }. We × shall show that e−iH t/h¯ in + can be defined only for t ≥ 0. This means that × e−iH t/h¯ does not form a one-parameter group (with −∞ < t < +∞) but only a semigroup, as was predicted in general in (56), (57) by the Paley–Wiener theorem [23]. To demonstrate this here explicitly for the Gamow ket, we make the substitution ψ − → eiH t/h¯ ψ − in (52) and obtain  +∞ 1 1 dE e−iEt/h¯ ψ − |E −  . (59) eiH t/h¯ ψ − |zR−  = − 2π i −∞I I E − zR Here we have used eiH t/h¯ ψ − |E −  = ψ − |E − e−iEt/h¯ . Provided that e−iEt/h¯ ψ − |E −  2 ∩ S, we can use the Titchmarsh theorem cf. (52) for the is an element of H− −iEt/h¯ 2 function G− (E) ≡ e ψ − |E −  iff this is a Hardy function ∈ H− . Since, for

298

A. BOHM

2 ψ − |E −  ∈ H− ∩ S, it follows that also 2 e−iEt/h¯ ψ − |E −  ∈ H− ∩S

if and only if t ≥ 0,

(60)

we can use the Titchmarsh theorem for the right-hand side of (59) only if t ≥ 0. Then, it follows that the right-hand side of (59) is equal to e−izR t/h¯ ψ − |zR− , and we obtain for all ψ − ∈ + : eiH t/h¯ ψ − |zR−  ≡ ψ − | (eiH t/h¯ )× |zR−  = e−izR t/h¯ ψ − |zR−  This we write as an equation in the dual space × + as −iH × t/h¯

(eiH t/h¯ )× |zR−  = e+

if and only if t ≥ 0. (61)

|zR−  = e−izR t/h¯ |zR−  

= e−iER t/h¯ e− 2 t/h¯ |zR− 

only for t ≥ 0.

(62)

Note that the first equality in (61) is the definition of the conjugate operator. The second equality is the result of the use of the Titchmarsh theorem to go from the right-hand side of (59) to the right-hand side of (61). In (62), the first equality is just a use of our new notation for the conjugate operator −iH × t/h¯

(eiH t/h¯ )× + ≡ e+

(63)

which we introduced in (59). The second equality in (62) is the result (61). A comparison of the second equality in (62) shows the consistency of the notation (61), −iH × t/h¯ |zR−  = e−izR t/h¯ |zR−  only for 0 ≤ t < ∞. (64) e+ 7.

Summary and conclusion The idea that there is a relation between a scattering resonance and an exponentially decaying state is not new. The scattering resonance is defined by a Breit–Wigner amplitude (35) and is observed in the cross section (36). An exponentially decaying state is observed as the exponential time dependence of decay rate (37a) and decay probability (37b). The exponential time dependence is also the property attributed to a Gamow vector (40). But it is well known [15] that such vectors with the property (39), (40) do not exist in the Hilbert space. It was also known that the relationship τ = h¯ /  between an exponential lifetime τ in (37b) and a width  in (36), could not hold exactly since standard techniques of theoretical physics using Wigner-Weisskopf approximation methods [14], led to the “addition terms” in (38). In contrast to the theoretical results of the conventional theory, the experimental evidence has substantially increased over the years, that Breit–Wigner resonances of energy and width (ER , ) are exponentially decaying states (ED , τ ) with (ED = ER , τ = h/ ¯ ). This is exemplified by the data (41). Also, the exponential decay law is one of the best established laws of nature [19], for the following reasons:

RESONANCES/DECAYING STATES

299

If the interference effects with the background and other nearby decaying states are properly eliminated from the experimental data a perfectly exponential decay rate results, see e.g. [18]. It is reasonable to attribute the exponential time dependence to the decaying state per se and deviations thereof to interference with other states. The Breit–Wigner amplitude (36)—and its relativistic version—has also been one of the most frequently used tools to analyze the resonance scattering data. Therefore, the following questions are justified: Can one modify the standard mathematical axioms of quantum theory such that Lorentzian amplitudes (35) and Gamow kets (39) are included as autonomous and fundamental ingredients of a new quantum theory15 ? And can one do this modification of the theory in such a way that τ = h¯ / ? In this paper, the following has been shown: One starts with a resonance pole at zR of the S-matrix element Sj (E) in (44). This pole term on the right-hand side of (46) is related by the Cauchy theorem to the Gamow ket-bra |zR− + zR | on the left-hand side of (46). To relate the Gamow kets to the Breit–Wigner (35) the energy wave functions φ + (E) = + E | φ +  of the in-state vector φ + (fulfilling the Schr¨odinger equation), and the energy wave functions ψ − (E) = − E |ψ −  of the out-“state” vector ψ − (fulfilling the Heisenberg equation), must obey the relation (47). This relation (47) means, that the Breit–Wigner on the r.h.s of (47) maps by integration the function ψ − | E − + E | φ +  to its value ψ − | zR− + zR | φ +  at E = zR = ER − i/2. This is the mathematical property which identifies the wave functions as Hardy functions. It suggests that the energy wave functions + E |φ +  = φ + (E) of the in-state vector φ + are Hardy function of the lower complex energy semi-plane (2nd sheet of the S-matrix) where the resonance pole zR is located. Similarly, the energy wave function − E |ψ −  of the out-observable ψ − must be a Hardy function of the upper complex plane.16 A mathematical property of the Hardy functions is that its values on the positive real axis (i.e. for physical values at scattering energy 0 ≤ E < ∞) already determine the function in the whole complex semi-plane. Since one also wants the Dirac basis vector expansion (28) and (29) to hold, the space of observables (or of detected out-“states”) {ψ − }, and the space of prepared in-state {φ + } is given by vector spaces whose energy wave functions are also Schwartz space functions. This means the energy functions must be smooth Hardy functions that can be analytically continued 15 There

exist only very few discrete energy eigenstates of the Hamiltonian with real eigenvalue En in nature, e.g. ground states of atoms and molecules and nuclei and a few relativistic elementary particles, most energy eigenstates or elementary particles are observed as decaying states or as scattering resonances. 16 Then the ψ − |E −  = ψ − (E) in (47) are Hardy functions of lower complex semi-plane and the energy E in the S-matrix element (34) can be continued into the lower complex energy plane where the resonance pole is located at zR = E − R − i/2.

300

A. BOHM

into the upper and lower complex energy plane, respectively. Precisely, the energy wave functions of the detected out-“states” = {ψ − } = +

2 are − E |ψ −  = ψ − (E) ∈ H+ ∩ S |R +

(65a)

and the energy wave functions of the prepared in-states = {φ + } = −

2 are + E |φ +  = φ + (E) ∈ H− ∩ S |R + .

(65b)

Therewith we have accomplished our goal: We have found the mathematical space for the prepared in-state {φ + }; it is given by the space of vectors  ∞  ∞ + + + + dE|E, j, j3 , η  E, j, j3 , η|φ  = dE|E + + E|φ +  (66) φ = j,j3 ,η

0

0

2 whose continuous coordinates + E | φ +  are the intersection of H− with + S(−∞, +∞) restricted to R+ . The {φ (t)} obey the Schr¨odinger equation. And we have found the mathematical space {ψ − } for the detected outobservables |ψ − ψ − | that obey the Heisenberg equation; it is given by the space of vectors  ∞  ∞ − − − − ψ = dE|E, j, j3 , η  E, j, j3 , η|ψ  = dE|E − − E|ψ −  j,j3 ,η

0

0

(67) 2 ∩ S(−∞, +∞)) |R+ . whose coordinates − E |ψ −  are in (H+ In their conjugate spaces there exist in addition to the complete set of basis kets − × |E ±  ∈ × ± with E ∈ R+ also other kets, e.g. the Gamow kets |zR  ∈ + of (53) with eigenvalue zR = ER − i/2 and with Breit–Wigner energy distribution given by 1 1 = with − ∞ < E < +∞. (68) E − zR E − (ER − i/2) The exponentially decaying Gamow ket | zR−  =| ER − i/2−  of the resonance position is associated to the Breit–Wigner resonance distribution 1 . E − (ER − i/2) This association between Gamow ket | zR−  and Breit–Wigner is given by the Titchmarsh theorem (52) for Hardy spaces, In the ket version, this relation is given by  +∞ 1 1 − dE |E −  , zR = ER − i/2, (69) |zR  = − 2π i −∞I I E − zR which is the Titchmarsh theorem (53), (54) for Hardy functions. This means in order to have a quantum theory that unifies Breit–Wigner resonances—on the right-hand side of (69)—and exponentially decaying states with

RESONANCES/DECAYING STATES

301

property (62) on the left-hand side of (69), the Hardy space axiom (65a,b) needs to hold. The time evolution, i.e. the solutions of the Heisenberg equation for observables ψ − ∈ + and the solutions of the Schr¨odinger for states φ + ∈ − , is now completely determined as a consequence of the Hardy space axiom: From the Paley-Wiener theorem for Hardy spaces follows that, in the Heisenberg picture, the solutions of the Heisenberg equation for the time evolved observables are given by t0 ≤ t < ∞. (70) ψ − (t) = eiH (t−t0 )/h¯ ψ − , Or, the solutions of the Schr¨odinger equation -in the Schr¨odinger picture- for the time evolved states are given by φ + (t) = e−iH

× (t−t )/h + 0 ¯φ ,

t0 ≤ t < ∞.

(71)

The time t0 is a finite value for which in mathematics one often chooses t0 = 0. This means the time evolution, in a quantum theory under the Hardy space axiom, is asymmetric, t0 ≤ t < ∞, and given according to (70) or (71) by a semigroup

U (t) = eiH (t−t0 )/h¯ with t0 ≤ t < ∞ for the observable detected vectors ψ − ∈ + or operators ,

(72)

or by the semigroup ×

U × (t) = e−iH (t−t0 )/h¯ with t0 ≤ t < ∞ for the prepared states φ + or state operators ρ .

(73)

Our conclusion therefore is. If one wants to have a quantum theory in which exponentially decaying states of lifetime τ and Breit-Wigner resonances of width  = h/τ are the same entities ¯ (for which there exists now substantial experimental evidence), then one is led to the Hardy space axiom (65). The states φ + obeying the Schr¨odinger equation (4) and observables |ψ − ψ − | obeying the Heisenberg equation (3), are represented by different Hardy subspaces (65) of the Schwartz space S of Dirac’s formalism. The time evolution (solution of the Schr¨odinger or Heisenberg equation) is then given by semigroups (72) and (73), respectively. This introduces a new entity into quantum theory, a beginning of time t0 (= 0) given by the semigroup time t0 (= 0) for the Hardy spaces evolutions (70) and (71). The physical meaning of time t0 needs some explanation, so that one does not think of t0 as a particular time in the life of the experimentalist. Consider a decaying state φ G ; quantum mechanics makes statements about ensembles of microphysical systems. The time t0 = 0 thus represents an ensemble of times [24]: t01 , t02 , . . . , t0n , . . . are the times when the 1-st, 2-nd, . . . , n-th, . . . microphysical system is produced; t11 , t12 , . . . , t1n , . . . are the times when 1-st, 2-nd,. . . , n-th,. . . micro-system decays at the decay vertex d1 , . . . , dn . The times t01 , t02 , . . . , t0n and the times t11 , t12 , . . . , t1n , . . .

302

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can be seconds or months or years apart. The time intervals tn = t1n − t0n during which the n-th micro particle lives and travels from the creation point at a target to the decay vertex, dn is the “proper lifetime of the n-th micro-system”. The average value of all tn is therefore the lifetime τ of the decaying state φ G (t). The time t = 0 for the decaying state vector φ G (t) is thus the class of times 1 {t0 , t02 , ..., t0n }; this is the “beginning of time” t0 = 0 for the decaying state vector φ G (t0 ). And it is this ensemble time tn = {t1n − t0n } which is represented by the time t in the Schr¨odinger or Heisenberg dynamical equation, and this time begins at the time t0 = 0 of the state preparation. To summarize: There is sufficient experimental evidence that scattering resonances (ER , , j ) and exponentially decaying states (ED , τ, j ) are the same quantum physical objects, ER = ED ,  = h/τ ¯ , associated to the first order pole of the S-matrix, Sj (E), of angular momentum j . A unified description of this observation leads to the pair of Hardy spaces. This Hardy space axiom predicts (by a mathematical theorem of Paley–Wiener) a semigroup time evolution and therewith a beginning of time, t0 = 0. Like all quantities of quantum physics, this t0 is observed as an ensemble t01 , t02 , t03 ,. . . for an ensemble of microphysical events [24], and the time intervals tn have been measured in quantum jump experiments on single, laser-cooled ions in a Paul trap [25]. Acknowledgement This paper was presented at the 42nd Symposium on Mathematical Physics, Toru´n, Poland. It is dedicated to Hellmut Baumgartel on the occasion of his 75th birthday. He was the first to refer us to the wonders of Hardy spaces. Help, support, and advice in the preparation of the manuscript from Peter Bryant and Hai V. Bui is gratefully acknowledged. We also acknowledge support from US NSF (OISE-0421936) while attending the conference. REFERENCES [1] A. Bohm, H. Uncu and S. Komy: Rep. Math. Phys. 64 (2009), 5. [2] M. H. Stone: Ann. Math. 33(3) (1932), 643; J. von Neumann: Ann. Math. 33(3), 567 (1932). [3] A. Bohm and M. Gadella: Dirac Kets, Gamow Vectors and Gelfand Triplets, Lecture Notes in Physics, vol. 348, Springer, Berlin 1989, proposition II p. 82; A. Bohm: Quantum Mechanics, 2nd, 3rd ed., Springer, Berlin 1986, 1993, Ch XXI; O. Civitarese and M. Gadella: Phys. Rep. 396, (2004), 41–113. [4] E. Merzbacher: Quantum Mechanics, Wiley, New York 1970 (Chapter 18). [5] R. P. Feynman: Rev. Mod. Phys. 20 (1948), 367. [6] T. D. Lee: Particle Physics and Introduction to Field Theory, Harwood Academic, New York 1981. [7] M. Gell-Mann and J. B. Hartle, in: J. J. Halliwell et al. (eds.), Physical Origins of Time Asymmetry, Cambridge University Press, Cambridge 1994; M. Gell-Mann and J. B. Hartle: UCSBTH-95-28, University of California at Santa Barbara, 1995; M. Gell-Mann and J. B. Harte: Proceedings of the 4th Drexel

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