Compendium of Quantum Physics || Rigged Hilbert Spaces and Time Asymmetric Quantum Theory

660 Rigged Hilbert Spaces and Time Asymmetric Quantum Theory

24. A. Bohm: J. Math. Phys. 22, 2813–823 (1981)25. M. Gadella: J. Math. Phys. 24, 1462–469 (1983)26. A. Bohm, M. Gadella: Dirac Kets, Gamow Vectors and Gelfand Triplets, Springer Lecture

Notes in Physics, vol. 348. Springer, New York (1989)27. A. Bohm: J. Math. Phys., 8, 1551–558 (1967), Appendix B28. E. P. Wigner: Z. Phys. 40, 492 (1927)29. E. P. Wigner: Z. Phys. 43, 642 (1927)30. E. P. Wigner: Z. Phys. 45, 602 (1927)31. F. Hund: Z. Phys. 43, 788 (1927)32. H. Weyl: The Theory of Groups and Quantum Mechanics, 2nd edn. (Dover, 1950), (First

edition, 1931)33. E. P. Wigner: Group Theory and Its Applications to Quantum Mechanics and Atomic Spectra,

1st edn., 1931. (Academic Press, New York, 1952)34. V. Bargmann: J. Math. Phys. 5, 862–68 (1964)35. F. Bruhat: Bull. Soc. Math. France 84, 97–205 (1956)36. B. Nagel: in Studies in Mathematical Physics (Proc. Istanbul 1970), pp. 135–54, ed. by A.O.

Barut (ed.) (Reidel, Dordrecht and Boston 1970)37. G. Lindblad, B. Nagel: Ann. Inst. H. Poincare 13, 27–56 (1970)38. In the relativistic case, the time evolution semigroup generalizes to a semigroup of spacetime

translations into the forward light cone, which is invariant under the group of orthochronousLorentz transformations. In this sense, semigroup time evolution can be viewed as an expres-sion of causality [39, 40]

39. A. Bohm, H. Kaldass, S. Wickramasekara: Fortschr. Phys. 51, 569–603 (2003)40. A. Bohm, H. Kaldass, S. Wickramasekara: Fortschr. Phys. 51, 604–34 (2003)

Rigged Hilbert Spaces and Time AsymmetricQuantum Theory

A. Bohm and N.L. Harshman

Rigged Hilbert Spaces and Dirac’s Bra-Ket Formalism

The rigged Hilbert space (RHS) is a triplet of linear topological spaces

� ⊂ H ⊂ �×, (1)

which is obtained from a linear space with scalar product � by completing it withrespect to three topologies. A topology τ specifies the definition of convergence,and when a space is completed with respect to a topology τ , the τ -limit elements ofCauchy sequences are adjoined to the space. For example, in (1) the space H is anabstract Hilbert space, i.e. it is the completion of � with respect to the topology τHgiven by the norm ||φ|| = √(φ, φ). The space � is the completion with respect to astronger topology τ� and the space �× is the space of continuous functionals on �.

Rigged Hilbert Spaces and Time Asymmetric Quantum Theory 661

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The linear space with scalar product � is the space that most physics texts andpapers call “the Hilbert space”. The space �× is the space one needs in order togive a mathematical meaning to the formalism that Dirac introduced in the first edi-tion (1930) of his book [1], and which he simplified in the third edition (1947) [2]� Dirac notation. The space H is the space that von Neumann introduced in hisHilbert space formulation of quantum mechanics in 1932 [3], where he remarkedthat Dirac’s formalism [1] is “scarcely surpassed in brevity and elegance” but “inno way satisfies the requirements of rigor.” An example of a Hilbert space, alsocalled a realization of the abstract Hilbert space, is the space of Lebesgue squareintegrable functions L2. Unfortunately, in this space one cannot define the scalarproduct of functions with the commonly used Riemann integrals; instead one mustuse Lebesgue integrals to obtain the complete Hilbert space L2 (and not just a real-ization of the linear space �).

Von Neumann’s Hilbert space provides a mathematically rigorous formulationof quantum mechanics, but it has some physically unintuitive features. For instance,a state is represented not by a single wave function, but by a class of Lebesguesquare integrable functions that differ from each other on a set of measure zero,which could even be the set of rational numbers. In contrast, physicists measureprobability distributions at only a finite number of points and then interpolate thedata with smooth functions. This practice suggests that states are better representedby functions φ(E) that have the following properties: they are continuous, infinitelydifferentiable, and they and their derivatives decrease for E → ∞ faster than anyinverse power of E. These properties define the Schwartz function space.

The standard example, used in quantum mechanics [4–6], group representa-tions [7–11] and axiomatic quantum field theory [12], is the following RHS (1):the space � is realized by the space of Schwartz functions on the positive real lineS(R+). The space H is realized by Lebesgue square-integrable functions L2(R+)and the space �× is realized by the space of tempered distributions S×(R+), whichincludes generalized functions like the Dirac delta defined below.

In quantum mechanical applications of RHSs, the space � is identified as thespace of physical states, i.e. those states that can be prepared and measured by ex-periments. The � observables that act as linear operators on the physical statesshould be represented by continuous linear operators in � and the set of these ob-servables are represented by an algebra of � operators. That means observables canbe multiplied and added without worrying about domain questions since they are alldefined everywhere in �. This feature is of enormous importance for practical cal-culations, but cannot be implemented in the Hilbert space even for the basic algebraof observables generated by position Q, momentum P , and energy H operators thatfulfill the Heisenberg commutation relations

(QP − PQ) = [Q,P ] = i� (2a)

and

H = P 2

2m+ V (Q). (2b)


Within the Dirac formalism it is tacitly assumed that observables can be addedand multiplied. This means that the space � must be constructed such that theobservables form an algebra of linear operators defined on the linear space ofstates �. (3)

Observables are measured by numerical values; therefore the operators that rep-resent them should have eigenvalues and eigenvectors. One can prove, however,that for certain operators, such as P and Q in (2a), there are no eigenvectors in theHilbert space. Nonetheless, Dirac postulated that the observables (like P , Q, and H

in(2a), but also more generally) have a complete set of eigenvectors, the Dirac kets.These kets were postulated to have the following two properties:

(i) On them, the observables have a set of eigenvalues that are discrete, continuous,or a combination of continuous and discrete:

H |E〉 = E|E〉, with 0 � E <∞ and/or E ∈ {E1, E2, ..., En, ...} (4a)

P |p〉 = p|p〉, with −∞ < p <∞ (4b)

Q|x〉 = x|x〉, with −∞ < x <∞. (4c)

This means the kets are labeled by the eigenvalues such as x, p, and E.(ii) These vectors provide a basis system and every vector ψ ∈ � can be uniquely

represented by a linear combination of these basis vectors.

As an example of the second point, consider the case that H has only a discreteset of eigenvectors |En〉. Then every ψ is expanded as

|ψ〉 =∑n

|En〉cn =∑n

|En〉〈En|ψ〉, (5a)

where the coordinates or components cn = 〈En|ψ〉 are complex numbers. The ba-sis vectors |En〉 are orthonormal (orthogonal and normalized) if H is self-adjoint(H = H †), i.e.

(|Ei〉, |Ej 〉) = 〈Ei |Ej 〉 = δij ={

1 i = j

0 i �= j. (5b)

The norm of every state vector ψ is finite and is calculated as

(ψ,ψ) =∑n

〈ψ|En〉〈En|ψ〉 =∑n

|〈En|ψ〉|2 =∑n

|cn|2 <∞. (5c)

This holds, for instance, if H in (2b) has the particular form of a quantum oscillatorwith mass m and spring constant k:

H = P 2

2m+ 1

2kQ2 (6)


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with En = �ω(n + 1/2) (n = 0, 1, . . .) and where ω = √k/m. The equations (5)are the infinite dimensional generalizations of the basis vector expansion of a threedimensional vector x =∑3

i=1 eixi .In general one cannot find for every self-adjoint operator such as H or P a com-

plete set of eigenvectors such that (5) holds. However, in the RHS (1) realized bythe Schwartz space

S(R) ⊂ L2(R) ⊂ S×(R), (7)

for every vector ψ ∈ � and every self-adjoint operator, the continuous analogues of(5a) hold:

|ψ〉 =∫ ∞

−∞dp |p〉〈p|ψ〉 =

∫ ∞

−∞dp |p〉ψ(p), (8a)

|ψ〉 =∫ ∞

−∞dx |x〉〈x|ψ〉 =

∫ ∞

−∞dx |x〉ψ(x), (8b)

|ψ〉 =∑n

|En〉〈En|ψ〉 +∫ ∞

0dE |E〉〈E|ψ〉=

∑n|En〉cn +

∫ ∞

0dE |E〉ψ(E) .(8c)

The kets |x〉, |p〉, and |E〉 exist as generalized eigenvectors of the operators Q,P , and H (or any other self-adjoint operator representing an observable). They areelements of �× and the eigenvalue equations (4) are defined to mean

〈ψ|H×|E〉 ≡ 〈Hψ|E〉 = E〈ψ|E〉 for all ψ ∈ �, (9)

and similarly for |p〉, |x〉, etc. The coordinates or wave functions 〈E|ψ〉 = 〈ψ|E〉∗are elements of the Schwartz space S(R), not of L2(R), and the generalized basisvector expansions (8) do not hold for every ψ ∈ H but only for every ψ ∈ �. Theoperator H×, called the conjugate operator of H , is defined generally for all linearcontinuous operators A on φ by

〈Aψ|F 〉 = 〈ψ|A×|F 〉 (10)

for all ψ ∈ � and all F ∈ �×. Since the space � is constructed such that thephysical observables are represented by continuous operators on the space �, theconjugate A× is a continuous operator in �×. The conjugate A× is an extensionof the Hilbert space adjoint A†: A† ⊂ A×. The observables form an algebra ofoperators in � as well as in �×. In contrast, in the Hilbert space H, one cannot havea continuous algebra of observables even for the canonical commutation relations(2a). For an example of how to construct the Schwartz space for the operator algebraof (2a) and (6) see [13], Sect. 1.

The continuous analogue of (5c) is now

(ψ,ψ) =∫ ∫

dE dE′〈ψ|E〉(|E〉, |E′〉)〈E′|ψ〉, (11)


but for this to make sense one needs the continuous analogue (|E〉, |E′〉) of theKronecker delta δnm of (5b). This is obtained if one takes the generalized scalarproduct (|E′〉, ψ) (precisely, the functional 〈E′| at any element ψ ∈ �):

〈E′|ψ〉 =∫ ∞

0dE 〈E′|E〉〈E|ψ〉. (12)

By treating the new symbol 〈E′|E〉 introduced in (12) as though it were a scalarproduct like (5b) but extended to continuous values, the object 〈E′|E〉 has the prop-erty that it maps the function ψ(E) ∈ S(R+) by integration over the positive realaxis to its specific value ψ(E′) at a particular energy E′. A mathematical objectwith such a property did not exist in the mathematics of the 1920s and 1930s, butonly achieved rigorous definition when Schwartz created the theory of distributionsor generalized functions 20 years later [14]. Dirac’s formalism was unhindered byall these mathematical complications. He postulated the properties (4) and (8), andsince (5b) held for the discrete case, he introduced the Dirac delta “function”

〈E′|E〉 = δ(E − E′) (13)

and stipulated that it fulfill (12). It is not truly a function, but it is a distribution andan element of S×(R+).

The requirements expressed in (3), (12), and (13) form the basis of Dirac’s for-malism for quantum mechanics. Inspired by this, first Schwartz created the theoryof distributions [14]. Then, extending this work, the Gel’fand school [15, 16] in-troduced into mathematics the RHS for the spectral analysis of � self-adjoint andunitary operators. Their nuclear spectral theorem is the mathematical version ofDirac’s continuous basis vector expansion (8). The RHS is the mathematical struc-ture in which various assumption of Dirac’s formalism, e.g. (2a), (4), (8c), (12), and(13), can be realized [4–11].

Thus, the Dirac formalism has been given a mathematical meaning by theSchwartz-RHS. The RHS’s of quantum physical systems are constructed such thatthe fundamental observables, like momentum, energy, and position (and many more,such as angular momentum and intrinsic observables like charges and isospin usu-ally connected with groups of transformations of space time and of charge spaces)are represented by an algebra of continuous operators. Then one chooses a completecommuting system of observables. For the oscillator this is just one operator, for ex-ample H , P or Q, and the Dirac basis vector expansion for the operator is like (5a),(8a), or (8b) for the oscillator. For other quantum systems, for example a particlein a spherically symmetric potential of the three dimensional space, the completesystem of commuting observables consists of three operators, either the momentumoperators P1, P2, and P3 or the Hamiltonian H and angular momentum operatorsJ2 and J3, and possibly some other set of observables that measure, for example,the internal properties and whose eigenvalues are collectively labeled as η. Then theDirac basis vector expansion (nuclear spectral theorem) is


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ψ =∑jj3η

∫dE |E, j, j3, η〉〈E, j, j3, η|ψ〉. (14)

The energy wave functions 〈E, j, j3, η|ψ〉 = ψjj3η(E) are Schwartz space func-tions if we use for the RHS the abstract Schwartz space.

Hardy Space Triplets for Resonance Scattering and Decay

The Schwartz-RHS gives a mathematical justification for the Dirac formalism. Itdefines the Dirac kets, justifies the algebraic manipulation of the observables, andproves the continuous basis vector expansion (8). However, it does not provide amathematical theory of scattering, resonances, and decaying states, and neither doesthe Hilbert space formulation. The description of resonances and decay phenomenain standard quantum mechanics is provided by the Weisskopf–Wigner approxima-tions [17, 18] and it is well-known to experts that “there does not exist...a rigoroustheory to which these various methods can be considered and approximation” [19].This is connected with the Stone–von Neumann theorem [20, 21] which states thatthe solutions of the � Schrodinger equation in H are given by the time-symmetricunitary group U†(t) = exp(−iHt) (or by the unitary group U(t) = exp(iHt)) forall times −∞ < t < ∞. In the RHS formulation using the Schwartz space, thespace � (and not H) is the set of physical states. That means one has to solve theSchrodinger equation

idφ(t)

dt= Hφ(t) (15)

under the boundary condition that φ ∈ �. Note that � and H have different defi-nitions of convergence, therefore the limits involved in taking the derivative of φ inthe space � is different from taking the limit in H. The τH-limit is defined by onenorm, whereas the τ�-limit is stronger and given by countably infinite number ofnorms. Thus the solutions of (15) in � do not have to be the same as the solutionsin H [22], but for the Schwartz-RHS, the solutions to (15) also have the group timeevolution property:

φ(t) = e−iHtφ, for all −∞ < t <∞ and for all φ ∈ �. (16)

Therefore, the Hilbert space axiom of standard (von Neumann) quantum theory,

set of physical states = {φ} = H = Hilbert space, (17)

as well as the Schwartz space axiom of the mathematical theory for the Dirac for-malism,

set of physical states = {φ} = � = abstract Schwartz space, (18)


lead to the same reversible time evolution. The time evolution of the prepared statesfulfills (16) and there will exist a state φ(t) for every t > 0 and also for every t < 0.

The physical quantities measured in experiments with quantum systems are theBorn probabilities. For instance, the probability to measure an observable & =|ψ〉〈ψ| in the state φ is given by the Born probability |〈ψ|φ(t)〉|2, and accordingto (16), this is predicted for every time −∞ < t < ∞. However, this contradictscausality; a quantum mechanical state must be prepared first at some time t0 beforethe observable can be measured in this state at times t > t0. That means Born proba-bilities |〈ψ|φ(t)〉|2 can be measured only for t > t0. Consequentially, the evolution(16) makes physical sense only for t � t0. In other words, instead of the unitarygroup solution, one should find solutions that obey semigroup evolution

φ(t) = e−iH(t−t0)φ, for only t > t0. (19)

Such solutions do not exist in the Schwartz space � or in the Hilbert space H.The time t0 before which “the state is defined completely by the preparation”

has already been mentioned by Feynman [23]. Gell-Mann and Hartle [24] appliedthis idea to the probabilities of histories for the expanding universe considered as aclosed quantum system. They did not derive (19); they restricted the time evolutionin (16) to t > t0 (where t0 is the time of the big bang) by fiat, violating the Hilbertspace and Schwartz space axioms (17) and (18). Other examples of systems with aphysically well-defined t0 are quasi-stable particles produced by the strong interac-tions that decay on a much slower time scale via the weak interaction [25]. That thedecay of excited atoms and of elementary particles is a time asymmetric (sometimesalso called irreversible) process has also been remarked in textbooks [26–28].

In the Hilbert space formulation of quantum mechanics [3], one cannot distin-guish between vectors φ describing states and vectors ψ describing observables like& = |ψ〉〈ψ| (or more general observables like A =∑n an|ψn〉〈ψn|). One assumesthat

set of states = {φ} = H = set of observable vectors = {ψ} (20)

and the time evolution for both is given by a unitary group for all times t . In the Diracformalism based on (18), one also identifies the set of state vectors and observablesvectors: {φ} = � = {ψ} (21)

and one has a single basis vector expansion such as (14) (or (8)) and one spaceof continuous antilinear functionals (kets) |E〉 = |E, j, j3, η〉 ∈ �×. In contrast,two sets of basis vectors are used in the heuristic conventional treatment of scatter-ing theory. These are the plane wave in-states |E+〉 and out-“states” |E−〉 that aresolutions of the Lippmann–Schwinger (LS) equation and are given by

|E±〉 = |E ± iε〉 = |E〉 + 1

E −H ± iεV |E〉, (22)

where ε → +0, (H − V )|E〉 = E|E〉, and V represents the scattering poten-tial [29–31]. The± iε in the LS equation (22) implies that the energy wave functions(ψ−(E))∗ = 〈−E|ψ−〉∗ = 〈ψ−|E−〉 and φ+(E) = 〈+E|φ+〉 are the boundary


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values of analytic functions in the lower complex energy semi-plane (for complexenergy z = (E + iε)∗ = E − iε, immediately below the real axis on the secondsheet of the S-matrix). In analogy to the Dirac expansion (8c), the |E±〉 are taken asbasis systems for the Dirac basis vector expansions

|φ+〉 =∫ ∞

0dE |E+〉〈+E|φ+〉, (23a)

and

|ψ−〉 =∫ ∞

0dE |E−〉〈−E|ψ−〉. (23b)

The ± iε in the phenomenological LS equation (22) suggests that the energy wavefunctions 〈−E|ψ−〉 are Schwartz functions that can be analytically continued intothe upper half complex energy plane (second sheet of the S-matrix) and the 〈+E|φ+〉are Schwartz functions analytic in the lower complex plane. Since the sets of vec-tors {φ+} and {ψ−} are defined by the sets of wave functions {〈+E|φ+〉} and{〈−E|ψ−〉}, it suggests that there are two RHS’s involved. One RHS

{φ+} = �− ⊂ H ⊂ �×− (24a)

is used for the set of state vectors {φ+} (in-states), which are defined by the prepa-ration apparatus, such as an accelerator. Another RHS

{ψ−} = �+ ⊂ H ⊂ �×+ (24b)

is used for the set of observable vectors {ψ−} (out-states, or better, out-observables),which are defined by the registration apparatus, such as a detector. The vectors φ+and ψ− are very similar to the in- and out-states in the S-matrix element of tradi-tional scattering theory [32–34]:

〈−ψ|φ+〉 = (ψ−, φ+) = (ψout, Sφin) = (ψout, φout). (25)

To specify the properties of the wave functions 〈−E|ψ−〉 and 〈+E|φ+〉 andtherewith the spaces �+ and �− of vectors ψ− and φ+, one checks under whichmathematical conditions on the spaces {〈−E|ψ−〉} and {〈+E|φ+〉} one can derivereasonable physical consequences from the hypothesis (24). A reasonable physicalconsequence would be a unification of resonance scattering and decay phenomena.One starts with the definition of a resonance by the S-matrix pole at the complexenergy zR = ER − i�/2. From the pole, one seeks the requirements that will allowthe derivation of two important signatures of time asymmetry: the Breit–Wigneramplitude for resonance scattering

aBWj (E) = Ri

E − (ER − i�/2)(26a)


and the exponentially decaying Gamow vector φG for the unstable states. TheGamow vector must be a ket φG = |(ER−i�/2)−〉 = |(ER−i�/2), j, j3, η

−〉 ∈ �×+(it is not in H, where exponentially decaying states are precluded [35]) with theeigenvalue property

〈Hψ−|(ER− i�/2)−〉≡〈ψ−|H×|(ER− i�/2)−〉=(ER− i�/2)〈ψ−|(ER− i�/2)−〉(26b)

for all ψ− ∈ �+. Here H× is the unique extension of H † = H to the space �×+.Further, φG must have the exponential semigroup evolution

〈e−iHtψ−|(ER − i�/2)−〉 ≡ 〈ψ−|e−iH×t |(ER − i�/2)−〉= eiERte−�t/2〈ψ−|(ER − i�/2)−〉 (26c)

for all ψ− ∈ �+ but only for t � 0 since the decaying state must first be preparedat a time t = t0 = 0 before it can decay.

The results (26a)–(26c) can be obtained if one assumes that in addition to beingSchwartz functions, the energy wave functions can be analytically continued intoeither the upper- or lower-half complex energy plane (second sheet) [36]. Precisely,the analytically continued wave functionsψ−(z) = 〈−z|ψ−〉 and φ+(z) = 〈+z|φ+〉are smooth Hardy functions 1 on the complex semiplanes C+ and C−, respectively:

φ+(E) = 〈+E|φ+〉 ∈ (H2− ∩ S)|R+ (27a)

ψ−(E) = 〈−E|ψ−〉 ∈ (H2+ ∩ S)|R+ . (27b)

The mismatch in signs between the wave function and the smooth Hardy spaces isfor historical reasons: the ‘±’ of the wave functions is the convention in scatteringtheory and has an independent origin from the ‘±’ of the Hardy space analyticityrequirements.

1 A precise definition of the smooth Hardy space is that a function ψ−(E) is in H2+ ∩ S if andonly if: (i) ψ−(E) belongs to the Schwartz space S , (ii) ψ−(E) admits analytic continuation,ψ−(z) = ψ−(E + iy), to the upper half plane (y > 0), and (iii) For any straight line in the upperhalf plane parallel to the real line, there exists a positive number K > 0 such that for all positivey > 0 the integral

∫∞−∞ |ψ−(E + iy)|2 dx < K is uniformly bounded by K , which means that the

bound is valid for a particular K and any y > 0. This integral is the usual Riemann integral andthe constant K depends on the specific function ψ−(E). The definition for H2− ∩ S is identical,just replacing the upper half plane by the lower half plane. Since any function in H2± ∩ S is ananalytic continuation of a function on the real line, it is automatically determined by its valueson any interval in the real line and viceversa. In particular, any function in H2± ∩ S is totallydetermined by its values on the positive half line and conversely. The spaces H2± ∩ S|R+ are thespaces of functions in H2±∩S , restricted to the positive semiaxis, i.e., in the functions in H2±∩S|R+ ,we have ignored their values on the negative part of the real line. This shows a one to one ontocorrespondence between the spaces H2± ∩ S and H2± ∩ S|R+ .


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With the pair of Hardy function spaces (27) one can construct a pair of Gel’fandtriplets of function spaces

(H2∓ ∩ S)|R+ ⊂ L2(R+) ⊂((H2∓ ∩ S)|R+

)×(28)

and show that these Hardy function spaces are locally convex nuclear spaces [37].Therefore, the Dirac basis vector expansions (23) are fulfilled as the nuclearspectral theorem for the Hardy space triplets (24). The time asymmetry (19) isthe mathematical consequence of the Paley–Wiener theorem [38] in the sameway the unitary group evolution is the consequence of the Stone–von Neumanntheorem [20, 21].

Therewith (27) is an axiom for the mathematical theory of quantum physics thatdistinguishes mathematically between prepared (in-)states described by the RHS(24a) and registered observables described by the RHS (24b) in the same way asthe experimentalists distinguish between the preparation apparatus of a state andthe detector of an observable. It provides a unified description of resonance anddecay phenomena and it leads to asymmetric, semigroup time evolution. Withoutthe mathematical notion of the RHS this time asymmetric quantum theory could nothave been conceived. See also � Time in quantum mechanics.

The authors would like to acknowledge fruitful discussion with Manuel Gadella.

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Compendium of Quantum Physics || Rigged Hilbert Spaces and Time Asymmetric Quantum Theory