Elements of a non-Hermitian quantum theory without Hermitian conjugation - scalar product and scattering

0.1 MB·English

by  Frieder Kleefeld

Elements of a non-Hermitian quantum theory without
∗
Hermitian conjugation — scalar product and scattering
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Frieder Kleefeld
1
0
2 Collaborator of the Centro de F´ısica das Interacc¸o˜es Fundamentais (CFIF),
n Instituto Superior T´ecnico (IST),
a Edif´ıcio Ciˆencia, Piso 3, Av. Rovisco Pais, P-1049-001LISBOA, Portugal
J Private permanent address:

6 Frankenstr. 3, 91452 Wilhermsdorf, Germany
e-Mail: [email protected], kleefeld@cfif.ist.utl.pt
]
h
p
-
t
n The descripitionofin a Hermitiansetting seeminglynonlocalandnon-
a perturbativephenomenalikeconfinementorsuperconductivityismostcon-
u
veniently performed by generalizing quantum theory to a non-Hermitian
q
regime where these phenomena appear perturbative and local. The short
[
presentationprovidesacluehowthiscanbedoneonthebasisofLorentzco-

1 variancewhile preservingthe analyticityofthe theory. After derivingwith
v the help of Lorentz covariance a quantum scalar product without making
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any use of metric or complex conjugation we sketch how the formalism of
4
scattering theory can be extended analytically to a non-Hermitian regime.
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1
0 PACS numbers: 03.65.Ca, 03.65.Nk, 03.65.-wi,11.30.-j, 11.55.-m, 11.80.Gw
.
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1. Introductory remarks
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1
Toustherearemainlythreepointsbecominggraduallyclear afterabout
:
v 20 years of intensive reseach effort to generalize quantum theory (QT) to a
i
X non-Hermitian (NH) setting (see e.g. Refs. [1]-[14] and references therein):
1)Thedescription ofphysicalsystemswithinan idealized Hermitian setting
r
a is at odds with experimental reality; 2) Various strong statements1 made
naively within a Hermitian setting do not hold in a NH setting; 3) The
∗ Partsofthecomprehensivepresentation“Stronginteractionswithquarksandmesons
— on unitarisation including bound states and non-Hermitian quantum theory” at
theworkshop on “UnquenchedHadron Spectroscopy: Non-PerturbativeModels and
Methods of QCD vs. Experiment (EEF70)” (1-5 September, 2014, University of
Coimbra, Coimbra, Portugal) (http://cfif.ist.utl.pt/∼rupp/EEF70/talks/Kleefeld.pdf).
1 E.g. that confinement cannot begenerated byscalar bosons or that thequarticcou-
pling of a Higgs scalar has — dueto stability reasons — to be positive [4].
(1)
2 kleefeld printed on January 8, 2015
advancedsectorofNHQTrequiredbyLorentzcovariance [2]andanalyticity
is in a Hermitian setting obtained by applying to the retarded sector a
Hermitian conjugation joint with a non-local, non-analytic metric [15].
2. Setup of non-Hermitian Quantum Theory (NHQT)
2.1. Covariance and conservation of complex energy in the complex plane
In the first place one should recall that the relativistic Klein-Gordon
equation being essentially the wave equation is a differential equation of
second order in the time coordinate which can be decomposed [2, 3, 6]
into two first order equations. For a — without loss of generality — time
independent eventually NH Hamilton operator H we have:
∂2 ∂ ∂
0 = (i¯h)2 H2 ψ(t) = i¯h H i¯h +H ψ(t) . (1)
∂t2 − !| i (cid:18) ∂t − (cid:19)(cid:18) ∂t (cid:19)| i
TherightsolutionoftheKlein-Gordonequation ψ(t) = ψ(+)(t) + ψ(−)(t)
| i | i | i
isthereforeobtainedbysuperimposingadditivelythesolutions ψ(+)(t) and
ψ(−)(t) of the retarded and advanced Schro¨dinger equation r|espectiviely:
| i
∂ ∂
0= i¯h H ψ(+)(t) , 0 = i¯h +H ψ(−)(t) . (2)
∂t − | i ∂t | i
(cid:18) (cid:19) (cid:18) (cid:19)
Therespective left eigen-solution ψ(+)(t) and ψ(−)(t) of these two equa-
tions is the right eigen-solution hhψ(+)(t))| hhψ(+)(t)|T and ψ(−)(t))
ψ(−)(t) T of the respective so-ca|lled “tran≡sphohsed reta|rded” a|nd “tran≡s-
hh |
posed advanced” Schro¨dinger equation (Here T denotes transpositon!), i.e.:
∂ ∂
0= i¯h HT ψ(+)(t)), 0 = i¯h +HT ψ(−)(t)). (3)
∂t − | ∂t |
(cid:18) (cid:19) (cid:18) (cid:19)
Inusingthenotationψ(±)(z,t) z ψ(±)(t) ((ψ(±)(t) z)T andψ(±)(z,t)
R L
((z ψ(±)(t)) ψ(±)(t) z T the≡nohnh-r|elativistiic≡one-dimen|sionallimitofEqs.≡
| ≡ hh | i
(2) and (3) reads in spatial representation:
∂ ¯h2 ∂2
(±) (±)
i¯h ψ (z,t) = +V(z) ψ (z,t), (4)
± ∂t R −2M ∂z2 ! R
∂ ¯h2 ∂2
(±) T (±)
i¯h ψ (z,t) = +V(z) ψ (z,t). (5)
± ∂t L −2M ∂z2 ! L
On the basis of these equations it is easy to show that there hold the fol-
lowing two continuity equations
∂ρ(+)(z,t) ∂j(+)(z,t) ∂ρ(−)(z,t) ∂j(−)(z,t)
= , = (6)
∂t − ∂z ∂t − ∂z
kleefeld printed on January 8, 2015 3
for the retarded and advanced energy densities ρ(+)(z,t) and ρ(−)(z,t) and
the respective energy current densities j(±)(z,t) defined as follows:
ρ(±)(z,t) = ψ(∓)(z,t)T ψ(±)(z,t) , (7)
L R
·
j(±)(z,t) =
1 ¯h2 ∂ψ(±)(z,t) ∂ψ(∓)(z,t)T ¯h2
(∓) T R L (±)
= ψ (z,t) ψ (z,t) .
L R
i¯h · 2M ∂z − ∂z · 2M !
±
(8)
The continuity equations (6) can be integrated along some suitable contour
connecting two points z and z in the complex z-plane yielding
1 2
∂ z2
dz ρ(±)(z,t) = j(±)(z ,t) j(±)(z ,t) . (9)
2 1
∂t Zz1 −(cid:16) − (cid:17)
Any integration contour with j(±)(z ,t) = j(±)(z ,t) defines an eventually
2 1
NHQT with a time-independent scalar product [1, 6]
z2 z2
dz ρ(±)(z,t) = dz ψ(∓)(z,t)T ψ(±)(z,t) = const (10)
L R
Zz1 Zz1 ·
replacing the well known scalar product of Max Born.
2.2. Elements of non-Hermitian scattering theory
Without loss of generality we consider now one-dimensional scattering
at a time-independent eventually NH potential V(z). For such a potential
the Schro¨dinger equations (4) and (5) can be solved by a separation ansatz
(±) (±) (±) (±)
ψ (z,t) = exp( Et/(i¯h))φ (z) and ψ (z,t) = exp( Et/(i¯h))φ (z)
R R L L
± ±
yielding the time-independent Schro¨dinger equations
¯h2 ∂2
(±) (±)
E φ (z) = +V(z) φ (z), (11)
R −2M ∂z2 ! R
¯h2 ∂2
(±) T (±)
E φ (z) = +V(z) φ (z), (12)
L −2M ∂z2 ! L
and according to Eqs. (8) the time-independent energy current densities
1 ¯h2 ∂φ(±)(z) ∂φ(∓)(z)T ¯h2
j(±)(z) = φ(∓)(z)T R L φ(±)(z) .
L R
i¯h · 2M ∂z − ∂z · 2M !
±
(13)
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Here we will discuss merely retarded scattering. Hence we usein the follow-
ing the abbreviations j(z) j(+)(z), φ(+)(z) φ(+)(z), φ(−)(z) φ(−)(z),
R L
≡ ≡ ≡
φ(+)(z)′ ∂φ(+)(z)/∂z and φ(−)(z)′ ∂φ(−)(z)/∂z. Advanced scattering
R L
≡ ≡
results are nonetheless easily derivable from their retarded counterparts.
In the region of vanishing potential, i.e. V(z) = 0, the solution of the
Schro¨dinger Eqs. (11) and (12) is of plane-wave type with k 2ME/¯h2:
0
≡
q
φ(±)(z) = exp( ik z) c(±)(k )+exp( ik z)c(±)( k ), (14)
0 0 0 0
± ∓ −
1 ¯h2 ¯h2
j(z) = φ(−)(z)T φ(+)(z)′ φ(−)(z)′T φ(+)(z)
⇒ i¯h · 2M − · 2M
(cid:16) ¯hk ¯hk (cid:17)
= c(−)(k )T 0 c(+)(k ) c(−)( k )T 0 c(+)( k ). (15)
0 0 0 0
· M − − · M −
In the following we want to consider retarded scattering between two points
z< and z> of vanishing potential, i.e. V(z<) = V(z>) = 0. Wave functions
and their derivatives at z> and z< are related by transfer matrices T(±):
2¯hM2 φ(±)(z>) = T1(1±) T1(2±) 2¯hM2 φ(±)(z<) , (16)
 q2¯hM2 φ(±)(z>)′   T2(1±) T2(2±)   q2¯hM2 φ(±)(z<)′ 
 q     q 
or, alternatively,
a(±) e(±) T˜(±) T˜(±) e(±)
1 = T˜(±) 1 = 11 12 1 , (17)
 e(±)   a(±)   T˜(±) T˜(±)  a(±) 
2 2 21 22 2
      
with
¯hk ¯hk
e(±) e±ik0z< 0 c(±)(k ) , e(±) e∓ik0z> 0 c(±)( k ) , (18)
1 ≡ s M < 0 2 ≡ s M > − 0
¯hk ¯hk
a(±) e±ik0z> 0 c(±)(k ) , a(±) e∓ik0z< 0 c(±)( k ) . (19)
1 ≡ s M > 0 2 ≡ s M < − 0
Simple algebra establishes the following relation between T˜(±) and T(±):
√k 1 1 1 1 1
T˜(±) = 1 + 0 ±ik0 (T(±) 1 ) . (20)
2 2 1 ∓ik10 ! − 2 ±ik0 ∓ik0 !√k0
1 is the2 2 unitmatrix. Moreover weassumetheenergy currentdensities
2
×
at points z< and z> to be equal, i.e. j(z<) = j(z>), yielding (see Eq. (15))
¯hk ¯hk
(−) T 0 (+) (−) T 0 (+)
c (k ) c (k ) c ( k ) c ( k ) =
> 0 > 0 > 0 > 0
· M − − · M −
¯hk ¯hk
(−) T 0 (+) (−) T 0 (+)
= c (k ) c (k ) c ( k ) c ( k ), (21)
< 0 < 0 < 0 < 0
· M − − · M −
kleefeld printed on January 8, 2015 5
(−)T (+) (−)T (+) (−)T (+) (−)T (+)
or, equivalently, a a e e = e e a a .
(−)T 1(+) · 1(−)T− 2(+) · 2(−)T 1(+) · 1(−)T− 2(+) · 2
Inspecting e e + e e = a a + a a we can
define the S-1matr·ix1S(+) an2d tr·ans2pose of1its in·ve1rse S(−2) = (·S(+2)−1)T by
(±) (±) (±) (±) (±)
a e S S e
1 = S(±) 1 = 11 12 1 . (22)
 (±)   (±)   (±) (±)  (±) 
a e S S e
2 2 21 22 2
      
Making use of Eqs. (17) and S(−)TS(+) = 1 we obtain:
2
S(±) S(±) T˜(∓)T −1 T˜(±)T˜(±)−1
S(±)= 11 12 = 11 12 22 .(23)
 S(±) S(±)   (cid:16)T˜(±)−(cid:17)1T˜(±) T˜(±)−1 
21 22 − 22 21 22
S(∓)T S(∓)T   T˜(±)−1 T˜(±)−1T˜(±) 
S(∓)T= 11 21 = 11 − 11 12 .(24)
 S(∓)T S(∓)T   T˜(±)T˜(±)−1 T˜(∓)T −1 
12 22 21 11 22
   (cid:16) (cid:17) 
The transmittivities T and T and reflectivities R and R are therefore:
1 2 1 2
T = 1 R = S(−)TS(+) = T˜(+)−1 T˜(−)T −1 = T˜(−)TT˜(+) −1, (25)
1 − 1 11 11 11 11 11 11
T = 1 R = S(−)TS(+) = T˜(−)T(cid:16)−1T˜(+(cid:17))−1 = (cid:16)T˜(+)T˜(−)T(cid:17)−1. (26)
2 − 2 22 22 22 22 22 22
(cid:16) (cid:17) (cid:16) (cid:17)
3. Simple application: scattering at a delta-potential
For the scattering at a delta-potential V(z) = g δ(z a) with g being
−
eventually complex-valued we choose x> = a + 0 and x< = a 0. The
−
delta-potential is represented by the following transfer matrices:
1 0 1 0
T(+) = , T(−) = . (27)
 2M g 2M 1   2M gT 2M 1 
¯h2 ¯h2 ¯h2 ¯h2
 q q   q q 
Invoking these transfer matrices into Eq. (20) we obtain
T˜(+) = T˜(−)T = S(−)T −1 = S(−)T −1 = 1+ 1 2M g 2M ,(28)
11 22 11 22 2i s¯h2k0 s¯h2k0
(cid:16) (cid:17) (cid:16) (cid:17)
T˜(−)T = T˜(+) = S(+) −1 = S(+) −1 = 1 1 2M g 2M ,(29)
11 22 11 22 − 2i s¯h2k0 s¯h2k0
(cid:16) (cid:17) (cid:16) (cid:17)
and, consequently,
T =T = 1 R =1 R =
1 2 1 2
− −
−1
1 2M 2M 1 2M 2M
= 1 g 1+ g . (30)
" − 2i s¯h2k0 s¯h2k0! 2i s¯h2k0 s¯h2k0!#
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Thisis—withoutinvolvinganycomplex conjugation —thestandardresult
which will be for one scattering channel obviously real-valued, positive and
within the invervall [0,1], if (Mg/k )2 is real-valued and non-negative.
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Acknowledgements
Best wishes to Everadus Johannes H.V. van Beveren (“Eef”) on the
occasion of his 70th birthday. Cordial thanks to him and his family for
lasting support. We also would like to deliver our best regards for kindest
hospitality to Maria do C´eu Martins Alfaiate Reste in the Rua da Bica 10,
3060-295 Porto dos Cov˜oes, Portugal, whereessential parts of the presented
work have been completed. This work was supported by the Funda¸c˜ao para
a Ciˆencia e a Tecnologia of the Minist´erio da Ciˆencia, Tecnologia e Ensino
Superior of Portugal, under contract CERN/FP/123576/2011 and by the
Czech project LC06002.
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