Quasi-hermitian Quantum Mechanics in Phase Space

0.56 MB·English

by  Thomas Curtright

UMTG - 3
Quasi-hermitian Quantum Mechanics in Phase Space
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2 Thomas Curtright§,†,* and Andrzej Veitia§,*

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J §Department of Physics, University of Miami, Coral Gables, Florida 33124

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1 †School of Natural Sciences, Institutefor Advanced Study,Princeton, New Jersey 08540

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February 1, 2008
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1 Abstract
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7 Weinvestigatequasi-hermitianquantummechanicsinphasespaceusingstandarddeformationquan-
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/ tization methods: Groenewold star productsand Wigner transforms. We focus on imaginary Liouville
h
p
theory as a representative example where exact results are easily obtained. We emphasize spatially
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n periodic solutions, compute various distribution functions and phase-space metrics, and explore the re-
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u lationships between them.
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:
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*[email protected] & [email protected]
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Contents
1 Introduction 4
2 Imaginary Liouville quantum mechanics 4
2.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Dual polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Phase space distributions 6
3.1 Eigenfunction WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Dual WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Wigner transform of the bilocal metric 8
4.1 Bilocal phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
↔
4.2 Liouville dual metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Homogeneous versus inhomogeneous ⋆genvalue equations 10
5.1 WFs as eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Entwining the dual metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 Inhomogeneities for dual WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.4 H from dual WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
h i
6 A sesquilinear star product and bracket 14
7 Other phase space distributions 15
7.1 Hybrid WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7.2 More hybrid WFs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
8 Direct solutions of the dual metric equation 21
9 The ⋆ root of the dual metric 21
9.1 S as a direct solution of an entwining equation . . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.2 The dual metric as an absolute ⋆ square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
e
9.3 S as a sum of hybrid WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
e
10 Meanwhile, back at the metric 25
10.1 R as a formal sum of the dual WFs (Not!) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2
10.2 Solving directly for R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
10.3 Solving for√⋆R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.4 √⋆R as a sum of hybrid WFs (Not!) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
10.5 Additional solutions for S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
11 Metric dependence of expectation values 33
11.1 Diagonal dual WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.2 Non-diagonal dual WFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.3 Expectations of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
12 Conclusions 40
References 41
Appendix A. The free particle limit 43
Appendix B. Arbitrary functions acting on WFs through star products 43
Appendix C. Conventions and star product compositions of bilinear Wigner transforms 46
Appendix D. Non-diagonal WFs, conventional and hybrid 47
Appendix E. A brief overview of biorthogonal systems and density operators 49
Appendix F. Operator expressions from Weyl transforms 53
3
1 Introduction
Superficially non-hermitian Hamiltonian quantum systems are of considerable current interest, especially in
the context of PT symmetric models [3, 13]. For such systems the Hilbert space structure is at first sight
verydifferentthanthatforhermitianHamiltoniansystemsinasmuchasthe dualwavefunctions arenotjust
the complex conjugates of the wave functions, or equivalently, the Hilbert space metric is not the usual one.
While it is possible to keep most of the compact Dirac notation in analyzing such systems (see Appendix
E), in the main body of this paper we will work with explicit functions and avoid abstract notation. Our
goal is to expose the underlying mechanisms (as in [8, 9]) rather than to hide them.
Our discussion is focussed on a system with potential exp(2ix). This model, as well as its field theory
extension,is ofinterestfor applications to table-topphysicalsystems [2, 4] andto deeper problems instring
theory [19, 20]. We will not discuss those applications here, but rather we will simply develop the phase-
space formalism for the point particle model. We believe this formalism will be helpful in understanding
the applications cited, as well as others. Other recent work along these same lines can be found in [16, 17].
2 Imaginary Liouville quantum mechanics
Consider“imaginary”or“periodic”Liouvillequantummechanicsasgovernedbytheapparently non-hermitian
Hamiltonian
H =p2+m2e2ix (1)
Without essential loss of generality, we take m = 1 in most of the following. Obviously, this is a “PT
symmetric” model. But more to the point, this is actually a “quasi-hermitian” theory [18] with a real
energy spectrum, as explained in [8] and as we shall clarify further here. We will analyze this system in
phase-space using the methods of deformation quantization.
2.1 Eigenfunctions
But first, let us briefly review the position representation Schr¨odinger eigenvalue problem for this system
(see e.g. [8]). With x on the real line and with the condition that the wave functions remain bounded, the
corresponding Schr¨odinger equation has energy eigenvalues given by all real E 0. The eigenfunctions are
≥
just Bessel functions, J eix . These are doubly degenerate when √E = n N, but they merge into
√E 6 ∈
±
a single, nondegenerate eigen(cid:0)fun(cid:1)ction when E = n2. For these nondegenerate cases the eigenfunctions are
4
2π-periodic in x, and the solutions are the analytic Bessel functions J [22].
n
d2
n2J eix = +e2ix J eix (2)
n −dx2 n
(cid:18) (cid:19)
(cid:0) (cid:1) 1 ∞ ( 1)k(cid:0) e(cid:1)2ikx
J eix = einx − , n=0,1,2, (3)
n 2n k!(k+n)! 4k ···
k=0
(cid:0) (cid:1) X
We willconsideronlysuchperiodicsolutionshere,since inouropinionthis is the mostinterestingsituation.
We note that these periodic eigenfunctions are superpositions of only right-moving plane waves. In fact,
in this periodic situation the discrete energy spectrum is precisely the same as would be found for particles
moving freely on a circle but restricted to non-negative momentum.
2.2 Dual polynomials
The periodic Bessel functions and their complex conjugates do not form an orthonormal set on the circle.
To obtain an orthonormal set of functions it is necessary to combine J (z) with an associated set of
n
{ }
polynomials in z 1, A (z) , the so-called Neumann polynomials. These are dual to J (z) on any
− n n
{ } { }
contour enclosing the origin z =0, in the following sense: 1 dz A (z)J (z)=δ . So on the circle, we
2πi z j k j,k
I
have
1 2π
dx A eix J eix =δ (4)
j k j,k
2π
Z0
(cid:0) (cid:1) (cid:0) (cid:1)
The Neumann polynomials on the circle are given explicitly by1
⌊n/2⌋(n k 1)!e2ikx
A0 eix =1 , A1 eix =2e−ix , An eix =2nne−inx −k!− 4k (5)
k=0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) X
d2
eix = +e2ix n2 A eix (6)
In −dx2 − n
(cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
As indicated, the A obey inhomogeneous modifications of Bessel’s equation, where the inhomogeneity is
n
{ }
either (z) z2 for even n or (z) z for odd n, according to
n n
I ∝ I ∝
ε e2ix for even n 0
eix = n ≥ (7)
n 
I
 2neix for odd n>0
(cid:0) (cid:1)
While(6)isinhomogeneous,neverthelesstheusualproofoforthogonalitybetweenpairsofnon-degenerateH
eigenfunctionsandtheirdualsgoesthroughbecausetheinhomogeneitiesareorthogonaltotheeigenfunctions.
2π
dx eix J eix =0 , for all j,k N (8)
j k
I ∈
Z0
(cid:0) (cid:1) (cid:0) (cid:1)
1Forconveniencewehavemodifiedtheusualnotationoftheassociatedpolynomialsasgivenin[1,15,22],namelyOn,and
havedefined An(z)=εn z On(z)whereε0=1andεn=2forn=0.
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5
3 Phase space distributions
We next compute Wigner transforms of various function bilinears for the 2π-periodic Bessel/Neumann sys-
tem. For a system so-defined, on a circle, momentum is quantized. Thus we would expect that the (x,p)
phase space is not the usual R2 nor even S1 R, but rather that it is reduced to S1 Z. In fact, the
× ×
periodic energy eigenfunctions of the imaginary Liouville Hamiltonian consist of superpositions of positive
momentum plane waves,so we would also expect not to need the Z momentum sector at all. Well, both
<0
expectations are almost true. But not quite. We shall see below to what extent these expectations are
born out.
3.1 Eigenfunction WFs
As a first step, we remind the reader about the structure of real (diagonal) Wigner functions (WFs) made
from 2π-periodic plane waves. They are just Kronecker deltas, with p Z as expected.
∈
1 2π
e (x,p) φ (x y) φ (x+y) e2iypdy (9)
n n n
≡ 2π −
Z0
=δ (10)
n,p
upon choosing φ (x) = exp(inx). By analogy, for the Liouville eigenfunctions ψ the WFs are again
n n
manifestly real, and again have support for p Z as expected, as given by
∈
1 2π
f (x,p) ψ (x y) ψ (x+y) e2iypdy (11)
n n n
≡ 2π −
Z0
( 1)p−n p−n e2ix(n−p+2k)
= − (12)
4p k!(n+k)!(p k)!(p k n)!
k=0 − − −
X
where the sum results from taking ψ (x) = J eix as given by the series in (3). Note for the Liouville
n n
case, as opposed to the free particle case, the su(cid:0)ppo(cid:1)rt in p is infinite: This particular f (x,p) is non-zero
n
for all p n 0. Another way to write these WFs makes use of the associated Legendre functions.2
≥ ≥
( 1)n isin2x p
f (x,p)= − LegendreP(p, n,icot2x) (13)
n
p!(p n)! 2 −
− (cid:18) (cid:19)
In particular, for any point in the reduced phase space, (x,p) S1 Z, we have
∈ ×
1 1
f (x,p)=1 δ (cos2x) δ + (2+cos4x) δ + (14a)
0 × p,0− 2 × p,1 25 × p,2 ···
1 1 1
f (x,p)= δ (cos2x) δ + (3+2cos4x) δ + (14b)
1 22 × p,1− 24 × p,2 283 × p,3 ···
1 1 1
f (x,p)= δ (cos2x) δ + (4+3cos4x) δ + (14c)
2 26 × p,2− 273 × p,3 21232 × p,4 ···
1 1 1
f (x,p)= δ (cos2x) δ + (5+4cos4x) δ + (14d)
3 2832 × p,3− 21132 × p,4 216325 × p,5 ···
2Aformwhichfacilitatesacontinuation tonon-integer p,shouldanyonewishtodothat.
6
Butnowwehavemorefunctionsatourdisposal,namelytheNeumannpolynomials,sowemaybuildanother
set of WFs for comparison to those in (10) and (12).
3.2 Dual WFs
For dual functions χ the WFs are also manifestly real, with p Z, as given by3
n
∈
1 2π
f (x,p) χ (x y) χ (x+y) e2iypdy (15)
n n n
≡ 2π −
Z0
f ⌊n/2⌋⌊n/2⌋(n k 1)!(n l 1)!
=4pn2 − − − − e2ix(l−k)δn p,k+l (16)
k! l! −
k=0 l=0
X X
where the sums result from taking χ (x)=A eix as given by the series in (5). Note the support in p is
n n
now finite: This particular f (x,p) is non-zero(cid:0)for(cid:1)0 n 2 n/2 p n. That is to say, 0 p n for
n
≤ − ⌊ ⌋≤ ≤ ≤ ≤
n even, and 1 p n for n odd.
f
≤ ≤
Resolving the constraint set by the Kronecker delta in (16) eliminates one sum and further restricts the
range of the other to yield4
min( n/2 ,n p)
⌊ ⌋ − (n k 1)!(k+p 1)!
fn(x,p)=4pn2 − − − e2i(n−p−2k)x (17)
k!(n p k)!
k=max(0,Xn−p−⌊n/2⌋) − −
f
In particular, again for any phase-space point (x,p) S1 Z,
∈ ×
f (x,p)=δ (18a)
0 p,0
fe1(x,p)=4δp,1 (18b)
fe2(x,p)=4 δp,0+32(cos2x) δp,1+64 δp,2 (18c)
× × ×
fe3(x,p)=36 δp,1+576(cos2x) δp,2+2304 δp,3 (18d)
× × ×
e
Thus,atanygivenmomentumlevel,wefindthesamesetoffunctionsofx(i.e. cos(2kx))nomatterwhether
we consider f or f .
n n
{ }
n o
There is a basic orthogonality relation for WFs and dual WFs.
f
1
f (x,p)f (x,p)=δ (19)
k n k,n
2π
Zx,p
P
f
This follows in a straightforward way from the bilinear structure of the Wigner transform and from the
orthogonality of the wave functions and their duals. There is also a corresponding pseudo-local relation on
3Todispelanyconfusionaboutourconventionsforfn,thecomplexconjugationin(15)isdifferent fromthatin(9)and(11)
f
justbecausewechosein[8,9]todefinethedualssuchthat δn,k = 21πR02πχn(x)ψk(x)dxwithoutanyexplicitconjugations.
4Thelimitsonthesummayalsobewrittenasmin( n/2 ,n p)andn p min( n/2 ,n p).
⌊ ⌋ − − − ⌊ ⌋ −
7
the phase space that involves the Groenewold star product (see (61) and (65) below). The form of these
results can be seen most easily through the use of formal density operator methods, as in Appendix E.
Perhapsitisusefultopresentthespecificexamplesoff andf ,forn=0,1,2,3andforp=0,1,2,3,in
n n
Tableform. Thisfacilitateschecking(19)forthese few cases,andilluminatesthe orthogonalitymechanism.
f
WFs & Dual WFs (non-zero values)
p=0 p=1 p=2 p=3
···
1 1 1 1 1
f ,f 1 , 1 cos2x , 0 cos4x+ , 0 cos6x cos2x , 0
0 0
−2 32 16 −1152 − 128 ···
e
1 1 1 1
f ,f , 4 cos2x , 0 cos4x+ , 0
1 1
4 −16 384 256 ···
e
1 1
f ,f 0 , 4 0 , 32cos2x , 64 cos2x , 0
2 2
64 −384 ···
e
1
f ,f 0 , 36 0 , 576cos2x , 2304
3 3
2304 ···
e
... ... ... ... ... ...
4 Wigner transform of the bilocal metric
It is explained in [8] – as well as in the classic literature on the subject – how a scalar product for a
biorthogonal system such as A ,J can always be written as an integral over a doubled configuration
k n
{ }
space involving a “bilocal metric” K(x,y).
(φ,ψ)= φ(x)K(x,y)ψ(y)dxdy (20)
ZZ
4.1 Bilocal phase space
↔
When a scalar product is so expressed as a bilocal bilinear form then it is naturally and very easily re-
expressed in phase space (which we suppose to be R2 in this paragraph) through the use of a Wigner
transform [11].
1 1 1
f (x,p) ψ x y φ x+ y eiypdy (21)
ψφ
≡ 2π − 2 2
Z (cid:18) (cid:19) (cid:18) (cid:19)
8
We have chosen the normalization here so that for p on the real line
∞
ψ(x)φ(x)= f (x,p)dp (22)
ψφ
Z−∞
More generally, Fourier inverting (21) gives the point-split product
φ(x)ψ(y)= ∞ ei(y x)pf x+y,p dp (23)
− ψφ
2
Z−∞ (cid:18) (cid:19)
Thus the scalar product (20) can be re-written as
(φ,ψ)= R(x,p)f (x,p)dxdp (24)
ψφ
ZZ
where the phase-space metric is the Wigner transform of the bilocal metric.
1 1
R(x,p)= eiypK x y,x+ y dy (25)
− 2 2
Z (cid:18) (cid:19)
and inversely
K(x,y)= 1 ∞ ei(x y)pR x+y,p dp (26)
−
2π 2
Z−∞ (cid:18) (cid:19)
In a more abstract notation (as in Appendix E) the form of (24) is φ,ψ =Tr(R ψ φ)=Tr ψ φ .
| ih | | ih |
(cid:16) (cid:17)
(cid:0) (cid:1)
f
4.2 Liouville dual metric
The preceding results are quite general. To be more specific, for 2π-periodic dual functions of imaginary
Liouville quantum mechanics, the scalar product was shown in [8] to be 1 2πdxdyχ (x)J(x,y)χ (y)
(2π)2 0 j k
=δ where RR
j,k
∞
J(x,y)=J e ix eiy =J e ix J eiy +2 J e ix J eiy (27)
0 − 0 − 0 n − n
−
n=1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) X (cid:0) (cid:1) (cid:0) (cid:1)
Again, just a Bessel function. Or, re-expressed in a form which is immediately useful in the following,
x+y
J(x,y)=J 2iei(y x)/2sin (28)
0 −
− 2
(cid:18) (cid:18) (cid:19)(cid:19)
Up to a normalization the corresponding metric in phase space is given by the Wigner transform of this
bilocal.5
A bit of care is needed since the Wigner transforms f (x,p) on which this metric will act are actually
n
defined so that a dual function χ plays the role of φ and a conjugate dual function χ plays the role of ψ in
f
the above (compare (21) to (15)). Thus, acting on f (x,p) the metric would be the Wigner transformof J
n
as above, only with first and second arguments interchanged. We also adjust the normalization here (and
f
5Totakethefreeparticlelimit,theparametermin(1)mustfirstberestored. SeeAppendixA.
9
again later, in (33)) to take into account our conventions and the fact that we are dealing with 2π-periodic
functions (see Appendix C). In view of all this, we finally obtain a dual phase-space metric given by
1 2π 1 2π
R(x,p)= J(x+w,x w)e2iwpdw = J0 2ie−iwsinx e2iwpdw
2π − 2π −
Z0 Z0
e 1 ∞ (sinx)2k 2π ∞ (sinx)2(cid:0)k (cid:1)
= e2iw(p k)dw = δ (29)
2π k=0 (k!)2 Z0 − k=0 (k!)2 p,k
X X
Hence the simple final answer.
sin2x p
R(x,p)= for integer p 0, but vanishes for integer p<0 (30)
(p!)2 ≥
(cid:0) (cid:1)
e
An equivalent operator expression can be obtained by the method of Weyl transforms. (See Appendix F.)
Another way to characterize R(x,p) is to note that it satisfies the differential-difference equation
e
p∂ R(x,p)=sin(2x)R(x,p 1) (31)
x
−
e e
even when p=0, since ∂ R(x,p=0)=0 as well as R(x,p= 1)=0. (This should be compared to (141)
x
−
given below. Note that R actually corresponds to R 1 in that later discussion.)
e −e
In fact, there is an obvious continuation of (30) to all real p, or even to complex x and p. Namely
e
sin2x p
R(x,p)= (32)
(Γ(p+1))2
(cid:0) (cid:1)
e
with its manifest zeroes and singularities (poles and cuts). This continuation also transparently satisfies
(31).
Taking into account all our conventions, we may now express the normalizations of pure states in terms
of the dual WFs and the dual phase-space metric as
1 1 n 2π sin2x p
ε = R(x,p)f (x,p)= f (x,p)dx (33)
n 2π Zx,p n 2π p=0Z0 (cid:0) (p!)2(cid:1) n
P X
e f f
where as usual ε = 1 and ε = 2 for n > 0. It is tedious but straightforward to use (17) to check this
0 n
normalizationand confirmthat the dual metric does its job. More importantly, (33) is consistentwith (19)
for the simple reason that
∞
R(x,p)= ε f (x,p) (34)
k k
k=0
X
This in turn follows from the expansion ofethe bilocal metric in terms of Bessel bilinears, in (27).
5 Homogeneous versus inhomogeneous ⋆genvalue equations
Were the dual functions just the complex conjugates of the wave functions, the two types of WFs that we
havedefinedwouldbeidentical,f (x,p)= f (x,p) ,butthisisobviouslynot trueforthecaseathand.
n n |ψ=χ
f
10