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2 Time Evolution, Partitioning Techniqueand Associated Dynamics

2 Time Evolution, Partitioning Techniqueand Associated Dynamics

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6



E.J. Brăandas



(t), respectively, we can separate out positive and negative times (with respect to

an arbitrary chosen time t = 0) as related with the relevant contours C± according to

G± (t) = ±(−i) θ (±t)e−iHt ; G(z) = (z − H)−1



(1.2)



where the retarded-advanced propagators G± (t) and the resolvents G(z± ); z± =

Rz ± iI z are connected through (R z, I z are the real and imaginary parts of

z = E + iε )

+∞

1

±

−izt

G(z)e dz; G(z) = G± (t)eizt dt

G (t) =







{t > 0; I z > 0}



−∞



{t < 0; I z < 0}



(1.3)



The Fourier-Laplace transform Eq. 1.2 exists under quite general conditions by e.g.

closing the contours C± in the lower and upper complex planes respectively. More

details regarding the specific choice of contours in actual cases can be found in

Refs. [13–15] and references therein. The formal retarded-advanced formulation

corresponding to Eq. 1.1 including the memory terms follows from

i





− H G± (t) = δ (t); ψ ± (t) = ±iG± (t)ψ ± (0)

∂t



i





− H ψ ± (t) = ±iδ (t)ψ ± (0)

∂t



(1.4)



and

(z − H)G(z) = I; ψ ± (z) = ±iG(z)ψ ± (0)

(z − H)ψ ±(z) = ±iψ ± (0)



(1.5)



ψ ± (t),



It is usual to normalize the time dependent wavefunction

which means that

ψ (z) obtained from partitioning technique as a result is not. Since we are primarily

interested in the case E ∈ σAC we will take the limits I (z) → ±0, obtaining the

dispersion relations

G(E + iε ) = lim (E + iε − H)−1 = P(E − H)−1 ± (−i)πδ (E − H)

ε→ ± 0



(1.6)



where P is the principal value of the integral.

The goal is now to evaluate full time dependence from available knowledge of the

wave function ϕ at time t = 0, for simplicity we assume that the limits t → ±0 are

the same, although this is not a necessary condition in general. Making the choice

O = |φ φ |φ −1 φ |; φ = ϕ (0), one obtains (the subspace, defined by the projector

O can easily be extended to additional dimensions)



ψ (0) = ψ + (0) = ψ − (0); Oψ (0) = ϕ (0); Pψ (0) = κ (0); O + P = I



(1.7)



1 Time Asymmetry and the Evolution of Physical Laws



7



Using familiar operator relations of the time dependent partitioning technique, see

again e.g. Ref. [15] for a recent review, we obtain

O(z − H)−1 = O(z − O H (z)O)−1 (I + HT (z))

H (z) = H + HT(z) H; T (z) = P(z − PHP)−1 P



(1.8)



As we have pointed out at several instances the present equations are essentially

analogous to the development of suitable master equations in statistical mechanics

[4–7], where the “wavefunction” here plays the role of suitable probability distributions. Note for instance the similarity between the reduced resolvent, based on

H (z), and the collision operator of the Prigogine subdynamics. The eigenvalues of

the latter define the spectral contributions corresponding to the projector that defines

the map of an arbitrary initial distribution onto a kinetic space obeying semigroup

evolution laws, for more details we refer to Ref. [6] and the following section.

Rewriting the inhomogeneous version of the Schrăodinger equation, where the

boldface wave vectors below signify added dimensions, one obtains (the poles of

the first line of Eq. 1.8 correspond to eigenvalues below)

(z − H)Ψ (z) = O(z − H (z))Oφ



(1.9)



From Eq. 1.9 the formulas of the time-dependent partitioning technique follows

straightforwardly



ϕ ± (t) = ±



i





O(z − H)−1 ψ (0) e−izt dz = ±





i





ϕ (z)e−izt dz



(1.10)







where



ϕ (z) = O(z − H)−1 ψ (0) = O(z − OH (z)O)−1 (ϕ (0) + HT(z)κ (0)



(1.11)



The equations of motion, restricted to subspace O, is directly obtained from the

convolution theorem of the Fourier-Laplace transform, i.e.

i





− OHO ϕ ± (t) = ±iδ (t)ϕ (0)

∂t







⎨ t

(GP (t − τ )PH ϕ (τ ))± d τ ± iG±

(t)κ (0)

+OHP

P





0



(1.12)

with

−iPHPt



=

P (t) = ±(−i)θ (t)e



1

2



T (z)eizt dz

C



(1.13)



8



E.J. Brăandas



and

t



(GP (t )PH ( ))± d τ = ±

0



i





T (z)H ϕ (z)e−izt dz



(1.14)







The first term on the right in Eq. 1.12 yields the description of the amplitude ϕ ±

evolving according to the Hamiltonian OHO. Furthermore the second term depends

on all times between 0 and t, while the last term evolves the unknown part κ at t = 0

completing the memory at initial time. Sofar no approximations have been set and

no loss of information acquiesced.

Introducing the auxiliary operator G±

L (t), through

+∞

±



±



iz t



dt

L (t)e



OH (z )O =

−∞



and the non-local operator G± (t; 0) via

t



(GL (t − τ )ϕ (τ ))± d τ = ±



±



G (t; 0)ϕ (0) =

0



i





OH (z± )Oϕ (z)e−izt dz (1.15)





one gets the more compact expression

i



∂ ±

ϕ (t) − G± (t; 0) ϕ (0) = ±i δ (t)ϕ (0) + OHPG±

P (t)κ (0)

∂t



(1.16)



Note that analogous evolution formulas hold within the subspace P, i.e. with

O and P interchanged. Although any localized wave packet under free evolution

disperses, it is however traditionally recognized that the complete formulation

of an elementary scattering set-up describes a time symmetric process provided

the generator of the evolution commutes with the time reversal operator and the

time-dependent equation imparts time symmetric boundary conditions. Compare

for instance analogous discussions in connection with the electromagnetic field,

obeying Maxwell’s equations, via retarded-, advanced- or symmetric potentials.

Although time symmetric equations may exhibit un-symmetric solutions via specific

initial conditions the fundamental point here concerns the “master evolution equation” itself. Hence, as already pointed out, we re-emphasize that no approximations

have been admitted and consequently time evolution proceeds without loss of

information.

At this junction it is common to discuss various short time expansions and/or

long time situations, i.e. to consider partitions of relevant time scales. We will

principally mention two interdependent scales, i.e. a global relaxation time τrel and

a local collision time τc . For instance, during τc the amplitude ϕ is not supposed to



1 Time Asymmetry and the Evolution of Physical Laws



9



alter very much. Hence one approximates the convolution in Eq. 1.16 for positive

times, i.e.

G+ (τc ; 0) ≈ OHO − iτc {OHPHO + 1/2!(−iτc)OH(PH)2 O · ·}



(1.17)



The relaxation time τrel obtains from time independent partitioning technique.

Accordingly starting from Eqs. 1.5 and 1.6 one attains the limits

(E − H)ψ ± (E) = ±iψ (0)

+∞

±



−1



ψ (E) = ±i lim (E + iε − H) ψ (0) = lim

ε →±0



ε →±0

−∞



ψ ± (t)ei(E+iε )t dt (1.18)



Incidentally we recover the stationary wave, ψc (E), via the causal propagator Gc (t)



ψc (t) = Gc (t)ψ (0) = ψ + (t) + ψ −(t); Gc (t) = e−iHt ;



(1.19)



and formally

+∞



ψc (E) = lim



ε →+0

−∞



+∞



ψ + (t)ei(E+iε )t dt + lim



ε →−0

−∞



+∞



ψ − (t)ei(E+iε )t dt =



ψc (t)eiEt dt.

−∞



Since the relaxation or life time, τrel , is directly related to a “hidden” complex

resonance eigenvalue of Eq. 1.9, or pole of Eq. 1.8, we need to derive and investigate

the dispersion relation for the reduced resolvent T (z), i.e.

lim T (E + iε ) = lim (E + iε − PHP)−1



ε →±0



ε →±0



= P(E − PHP)−1 ± (−i)πδ (E − PHP)



(1.20)



For instance, in the none-degenerate case one finds by getting the real and imaginary

parts of f (z)

f (z) = φ |H (z)|φ



(1.21)



f ± (E) = fR (E) ± (−i) f1 (E)



(1.22)



and



with

fR (E) = E + φ |HP(E − PHP)−1 H|φ

fI (E) = π φ |H δ (E − PHP)H|φ ≥ 0



(1.23)



In the limit ε → ±0, assuming full information for simplicity at the initial time,

t = 0, i.e. κ (0) = 0; φ = ϕ (0) = ψ (0), Eq. 1.9 yields

(E − H)Ψ ± (E) = fI (E) (0)



(1.24)



10



E.J. Brăandas



keeping in mind that z = E − iε from Eq. 1.23

E = fR (E); ε (E) = f1 (E)



(1.25)



Note that Eq. 1.25 only gives the resonance approximately. The exact complex

resonance eigenvalue (if it exists, see more on this below) has to be found by

analytical continuation, Eq. 1.9, by e.g. successive iterations until convergence for

each individual resonance eigenvalue. Hence in the first iteration, one gets the

lifetime τrel given by

f1 (E) = ε (E) = {2τrel (E)}−1



(1.26)



To find an “uncertainty-like” relation between the two time scales we combine the

expansion Eq. 1.17 with

OH (z)O = OHO +



1

1

OHPHO + OH(PH)2 O · ·

z

z



and

OH (E ± i0)O = OHO + OHPP (E − PHP)−1 PHO

±(−i)π OHPδ (E − PHP)PHO

obtaining



τc σ 2 (E) = ε (E) =



1

2 τrel



(1.27)



with σ2 being the variance at E0 = ϕ (0)|H|ϕ (0) , i.e.



σ 2 = ϕ (0)|HPH|ϕ (0) = ϕ (0)|(H 2 − ϕ (0)|H|ϕ (0) 2 )|ϕ (0)



(1.28)



The relations above contain many approximations, e.g. break-up of convolutions,

truncation of various expansions etc., not to mention the assumed existence of a

rigorous analytical continuation into higher order Riemann sheets of the complex

energy plane. Since the original Schrăodinger equation, or Liouville equation, as

pointed out, is time reversible and rests on a unitary time evolution, it is obvious that

the present lifetime analysis is contradictory. This is a well-known fact amongst the

practitioners in the field; nevertheless, a lot of physical and chemical interpretations

have been made and found to be meaningful portrayals of fundamental experimental

situations. How come that this still appears to work satisfactorily? In the next section

we will examine the reasons why, as well as develop the necessary mathematical

machinery to rigorously extend resolvent- and propagator domains and examine its

evolutionary consequences.

The object of our description is twofold: first to show that analytic continuation

into the complex plane can be rigorously carried out and secondly to examine



1 Time Asymmetry and the Evolution of Physical Laws



11



the end result for the associated time evolution. Not only will we find that time

symmetry is by necessity broken, but also that novel complex structures appear

with fundamental consequences for the validity of the second law as well as giving

further guidelines on a proper self-referential approach to the theory of gravity.



1.3 Non-self-Adjoint Problems and Dissipative Dynamics

Staying within our original focus in the introduction, we recapitulate the current

dilemma, i.e. how to connect, if possible, the exact microscopic time reversible

dynamics, see above, with a time irreversible macroscopic entropic formulation

without making use of any approximations whatsoever. In the present setting one

must ascertain precisely what it takes to go from a stationary- to a quasi-stationary

scenario. To begin with, we return to the practical problem of extracting meaningful

life times out of the exact dynamics presented above and in particular to dwell on

the consequences if any.

The understanding, interpretation and practical tools to approach the problem of

resonances states in quantum chemistry and molecular physics are basically very

well studied. Generally one has either (i) concentrated on the properties of the

stationary time-independent scattering solution (ii) attempted to extract the Gamow

wave by analytic continuation and/or (iii) considered the time-dependent problem

via a suitably prepared reference function or wave-packet. In each case the analysis

prompts different explanations, numerical techniques and understanding, see e.g.

Ref. [15] for a review and more details.

In order to appreciate the significance of this situation, we will portray one of the

most significant and successful approaches to quasi-stationary unstable quantum

states by re-connecting with the previously mentioned theorem due to Balslev

and Combes [3]. The authors derived general spectral theorems of many-body

Schrăodinger operators, employing rigorous mathematical properties of so-called

dilatation analytic interactions (with the absence of singularly continuous spectra).

The possibility to “move” or rotate the absolutely continuous spectrum, σAC ,

appealed almost instantly and was right away exploited in a variety of quantum

theoretical applications in both quantum chemistry and nuclear physics [16].

The principle idea stems from a suitable change, or scaling, of all the coordinates

in the second order partial differential equation (Schrăodinger equation), which if

allowing a complex scale factor, permits outgoing growing exponential solutions,

so-called Gamow waves, to be treated via stable numerical methods without being

forced to leave Hilbert space. Although this trick admits standard usage of alleged

L2 techniques, there is a price, i.e. the emergence of non-self-adjoint operators

which brings about a lot of important consequences to be summarized further below.

The strategy is best illustrated by considering a typical matrix element of a

general quantum mechanical operator W (r) over the basis functions, ϕ (r) and

φ (r), where we write r = r1 , r2 , . . . rN ; assuming 3N fermionic degrees of freedom.



12



E.J. Brăandas



Employing the scaling r = 3N r; η = eiϑ (or η = |η |eiϑ ), where the phase ϑ ≤ ϑ0

for some ϑ0 that in general depends on the operator, one finds straightforwardly



ϕ ∗ (r)W (r)φ (r)dr =







ϕ ∗ (r )W (r )φ (r )dr



(1.29)



or in terms Dirac bra-kets (with ϕ ∗ (η ∗ ) = ϕ (η ))



ϕ |W |ϕ = ϕ (η ∗ )|W (η )|φ (η )



(1.30)



We assume that the operator W (r) as well as ϕ (r) and φ (r) are properly defined

for the scaling process to be justified. For simplicity we also take the interval of the

radial components of r to be (0, ∞). Note that Eq. 1.29 contains the requirement that

the matrix element should be analytic in the parameter η , demanding the complex

conjugate of η in the “bra” side of Eq. 1.30. This is the reason why many complex

scaling treatments in quantum chemistry are implemented using complex symmetric

forms.

In order to appreciate the fine points in this analysis, we therefore return to the

domain issues, i.e. how to define the operator and the basis functions so that the

scaling operation above becomes meaningful. Following Balslev and Combes [3],

we introduce the N-body (molecular) Hamiltonian as H = T + V , where T is

the kinetic energy operator and V is the (dilatation analytic) interaction potential

(expressed as sum of two-body potentials Vij bounded relative Tij = Δij , where the

indices i and j refers to particles i and j respectively). As a first crucial point we

realize that the complex scaling transformation is unbounded, which necessitates a

restriction of the domain of H; note that H is normally bounded from below. Hence

we need to specify the domain D(H) of H as

D(H) = {Φ ∈ h, H Φ ∈ h}



(1.31)



where h denotes the well-known Hilbert space. The essential property of a dilatation

analytic operator is that each individual pair potential of the interaction V is bounded

relative the corresponding part of the kinetic energy. Hence the unboundedness is

due to the latter i.e. D(H) = D(T ). With these preliminaries one can prove that the

scaling operator U(ϑ ) = exp(iAθ ) is unitary for real ϑ and generated by

A=



1 k=N

∑ [pk xk + xk pk ]

2 k=1



(1.32)



where xk and pk are coordinate and momentum vectors of the particle k. As a result

we get

U(eϑ )Φ(r) = exp(iAθ )Φ(r) = e



3Nϑ

2



Φ(eϑ r)



(1.33)



and with ϑ → iϑ ; η = eiϑ , or more generally η = |η |eiϑ we write

H(η ) = U(η )H(1)U −1(η ) = η −2 T (1) + V (η )



(1.34)



1 Time Asymmetry and the Evolution of Physical Laws



13



At this stage it is crucial to emphasize that the formal expression Eq. 1.34 must

be obtained in two steps due to the unboundedness of T and the complex scaling

transformation. First we introduce Ω = {η , | arg (η )| ≤ ϑ0 } in agreement what

has been said above, then decompose Ω in its upper and lower parts, partitioned

by the real axis R, where Ω = Ω+ ∪ Ω− ∪ R and R = R+ ∪ R− ∪ {0}. To avoid

problems we will exclude the point {0}. The first step consists of the real scaling, i.e.

η ∈ R+ , which corresponds to a unitary transformation, followed by an analytical

continuation to η ∈ Ω+ , corresponding to a similarity, non-unitary operation. Since

this is an important point we will consider the scaling operator U(η ); η ∈ Ω+ more

exactly by bringing in the dense subset N (Ω)

N (Ω) = {Φ, Φ ∈ h; H(η )Φ ∈ h; U(η ) ∈ h; η ∈ Ω}



(1.35)



as the well-known Nelson’s class of dilatation analytic vectors [17] more specifically

defined as follows. A vector φ ∈ D(A) is an analytic vector of A if the series

expansion of eAϑ φ has a positive radius of absolute convergence, i.e.









n=0



An φ

n!



ϑn < ∞



for some ϑ > 0. For our purpose, to be explained below, we introduce the Hilbert

(or Banach) space norm

ϑ0



sup



η ∈Ω



U(η )φ < ∞;



U(η )φ



2

L2



dϑ = φ



2

N (ϑ0 )



(1.36)



−ϑ0



To include the kinetic energy operator in our discussion it is natural to request that

the first and second partial derivatives should also satisfy Eq. 1.36 hence introducing

(i)

the spaces Nϑ0 with i = 0, 1, 2 analogously.

With these preliminaries we can now make the precise definition of the selfadjoint analytic family H(η ) as

H(η ) = U(η )H(η )U −1 (η ); D(UHU −1 ) = Nϑ20

H(η ) = η −2 T (1) + V(η ); D(UHU −1 ) → D(T )



(1.37)



Recapitulating, the first step consists of restricting the Hilbert space to a smaller

(2)

domain Nϑ0 for which the scaling U is defined for all complex η values with its

arguments smaller in absolute value than ϑ0 . The second step, after the parameter

ϑ has been made complex (ϑ → i ϑ ), consists of completing the Nelson class of

dilation analytic vectors to the domain of H or in this case T . Here this means

convergence with respect to the standard L2 norm (for both the functions and its

first and second partial derivatives).



14



E.J. Brăandas



To appreciate the reason for our painstaking carefulness at this particular stage

we come back to our frequent references to the fundamental dilemma expressed

above and in the introduction. First the theorem of Balslev and Combes provides us

with a rigorous path into the second Riemann sheet of the complex energy plane.

The factor, η −2 , appearing in front of the kinetic energy operator T , see Eqs. 1.34

and 1.37, has a simple and natural effect. It means that the absolutely continuous

spectrum of H(η ) is rotated in the complex plane with a phase angle equal to −2ϑ .

In this process complex resonance eigenvalues become “exposed” in agreement with

the aforesaid generalized mathematical spectral theorem [3]. However, there is a

small price to be paid, viz, in the process of the analytic continuation the two steps

mentioned above entails a small inevitable loss of information represented by the

restrictions necessary for the definition of the whole analytic family of the operators

H(η ). As we will see this will have consequences both for the entropic as well as the

temporal evolution. These results, all the same, guarantee that the approximations

made in the previous section could be meaningful despite our words of warning.

There are in effect two principal consequences that we will examine. Firstly

the spectral generalization [3] in terms of appearing complex poles of the actual

resolvent, with the complex part interpreted essentially as the reciprocal life-time of

the state and secondly the dynamical outcome regarding the time evolution, i.e. the

conversion of an isometry to a contractive semigroup [18].

To appreciate the first generalization, i.e. modifying the projection operator

formulations of Sect. 1.2, the following construal is supplied

T (η ; z) = P(η )(z − P(η )U(η ) H(1)U −1 (η )P (η ))−1 P(η )

O(η ) = |φ (η ) φ (η ∗ ) |; P(η ) = I − O(η )



φ (η ∗ ) |φ (η ) = φ (1) |φ (1) = 1



(1.38)



Apart from giving the impression of being a rather formal extension, there are two

important points to consider. First the projectors are oblique, i.e. idempotent

O2 (η ) = O(η )

but not self-adjoint

O† (η ) = O(η ∗ ) = O(η )

Furthermore the present bi-orthogonal construction authorize non-probabilistic

formulations allowing e.g. the possibility of zero norms, viz. from Eq. 1.9 one may

encounter, starting with a none-degenerate eigenvalue, that



Ψ (η ∗ ; z∗ ) |Ψ (η ; z) = 1 + Δ(η ; z) = 1 − f (z) = 0



(1.39)



The emerging singularity is associated with a degeneracy of so-called Jordan-block

type, an abysmal situation in matrix theory; see e.g. Ref. [18] and references therein.

In our case, as we will see, this will actually be a “blessing in disguise.” In passing



1 Time Asymmetry and the Evolution of Physical Laws



15



we note that the observed loss of information carries an unexpected increase of

entropy. The occurrence of Jordan blocks, see more below about associated spectral

degeneracies and their interpretations, implies that full information as to the given

state becomes uncertain at the “bifurcation point”, with an associated entropic

increase as a result.

The second consequence regards the dynamics. As already pointed out the stepwise approach is of basic relevance for the use of dilatation analytic Hamiltonians

as generators of contractive semigroups. The problem of comparing classical and

quantum dynamics and the appropriate choice of so-called Lyapunov converters

were examined in some detail in Ref. [8]. Briefly we will review the implication

as follows. Consider an isometric semigroup, cf. the causal propagator in (1.19),

G(t);t ≥ 0, defined on some Hilbert space h. If there exists a contractive semigroup

S(t);t ≥ 0 and a densely defined closed invertible linear operator Λ, with the domain

D (Λ) and range R (Λ) both dense in h, such that

S(t) = ΛG(t)Λ−1 ; t ≥ 0



(1.40)



on a dense linear subset of h, then Λ is called a Lyapunov converter. A necessary

condition for the existence of Λ for a given G(t) = e−iHt is that the generator H has a

non-void absolutely continuous spectral part, i.e. σAC = ∅. In view of what has been

said above it is natural to ask whether H(η ) generates a contractive semi-group, i.e.

that (note that the + -sign in S+ is not a “dagger”)

S+ (t, η ) = U(η )G(t)U −1 (η ) = e−iH(η )t ; t ≥ 0



(1.41)



This is indeed true for many types of potentials, but unfortunately not for the case

of the attractive Coulomb interaction. Although the Balslev-Combes theorem for

dilation analytic Hamiltonians guarantee that the modified spectrum lies on the

real axis (bound states) and in a subset of the closed lower complex halfplane,

a further requirement (using the Hille-Yosida theorem) is that the numerical range

must also be contained in the lower part of the complex energy plane. In addition

the 1/r potential is problematic both at the origin and at infinity; see e.g. Ref. [19]

for a detailed treatment of resonance trajectories and spectral concentration for a

short-range perturbation resting on a Coulomb background. Here a resonance in

the continuous spectrum carries typical ground-state properties [20] and allows for

complex curve crossings (Jordan blocks) [21]. Since the long-range Coulomb part

in a many-body system will be screened by the other particles the anomalies of the

Coulomb problem should not be crucial with respect to the isometric-contractive

semi-group conversion in realistic physical systems. For additional discussions on

this point, involving a slightly more general definition in terms of quasi-isometries

see Ref. [8].

Summarizing; the loss of information, i.e. restricting the full unitary time

evolution to an isometry (weak convergence) before the conversion via a suitable

Lyapunov converter to a contractive semi-group (strong convergence) is an objective

process in contrast to the subjective preparation of any initial state involving various



16



E.J. Brăandas



levels of course graining. It is also important to realize that the completion of a

dense subset of Hilbert space with respect to the appropriate norm gives different

limits depending on whether it is carried out before or after the conversion, hence

we will speak of an informity rule, i.e. a certain natural loss of information, which

is compatible with broken temporal symmetry.

Finally, on account of the impairment of information loss, it is all the same

important to mention that, in the classical as well as in the quantum case, it is

not enough to conclude that the mere existence of a Lyapunov converter explains

or derives time irreversibility and, in the Liouville formulation, guarantees the

approach to equilibrium [8]. In the next section we will concentrate on the

degenerate state before moving on to the relativistic situation looking for the only

remaining explanation of irreversibility in terms of a formulation involving a spatiotemporal dependent background.



1.4 The Jordan Block and the Coherent Dissipative Ensemble

As already mentioned the informity rule prompts several consequences one being

the emergence of so-called Jordan blocks or exceptional points. Although belonging

to standard practise in linear algebra formulations we will proffer some extra time

to this concept. In addition to demonstrate its simple nature we will also establish

a simple complex symmetric form not previously obtained, see e.g. Refs. [11, 14,

21, 22]. Let us start with the 2 × 2 case, where it is easy to demonstrate that the

Jordan canonical form J and the complex symmetric form Q are unitarily connected

through the transformation B, i.e.

Q = B−1 JB = B† JB



(1.42)



where

Q=



1

2



1 −i

−i −1



; J=



0 1

0 0



1

; B= √

2



1

i

1 −i



(1.43)



We note that the squares of Q and J are zero, yet the rank is one. Although these

Jordan forms do not appear in conventional quantum mechanical energy variation

calculations they are not uncommon in extended formulations. For various examples

of the latter instigated in quantum physical situations, we refer to [11] and also to

applications of a new reformulation of the celebrated Găodel(s) theorem(s) in terms

of exceptional points, Ref. [12]. Note that the alternative formulation in terms of an

antisymmetric construction is not anti-hermitean as wrongly indicated in Ref. [12].

As demonstrated, complex symmetric forms are naturally exploited in quantum

chemistry and molecular physics and therefore we need to extend Eqs. 1.42 and

1.43 to general n × n matrices. The mathematical theorem that a triangular matrix is

similar to a complex symmetric form goes back to Gantmacher [23], but the explicit



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