QT-Symmetry and Weak Pseudo-Hermiticity

0.16 MB·

by  Ali Mostafazadeh

-Symmetry and Weak Pseudo-Hermiticity
QT
Ali Mostafazadeh
8
0
0 Department of Mathematics, Koc¸ University,
2
n 34450 Sariyer, Istanbul, Turkey
a
J [email protected]

8
2

] Abstract
h
p
- For an invertible (bounded) linear operator acting in a Hilbert space , we consider
nt Q H
the consequences of the -symmetry of a non-Hermitian Hamiltonian H : where
a
QT H → H
u is the time-reversal operator. If H is symmetric in the sense that H† = H, then -
q T T T QT
[ symmetryisequivalentto −1-weak-pseudo-Hermiticity. Butingeneralthisequivalencedoes
Q

2 nothold. Weshowthisusingsomespecificexamples. Amongtheseisalargeclassofnon- -
PT
v
symmetric Hamiltonians that share the spectral properties of -symmetric Hamiltonians.
9 PT
7
8
PACS number: 03.65.-w
4
.
0 Keywords: Antilinear operator, symmetry, -symmetry, Pseudo-Hermiticity, periodic po-
1 PT
7 tential.
0
:
v
i 1 Introduction
X
r
a
Among the motivations for the study of the -symmetric quantum mechanics is the argument
PT
that -symmetry is a more physical condition than Hermiticity because -symmetry refers to
PT PT
“space-time reflection symmetry” whereas Hermiticity is “a mathematical condition whose physical
basis is somewhat remote and obscure” [1]. This statement is based on the assumption that the
operators and continue to keep their standard meanings, as parity (space)-reflection and
P T
time-reversal operators, also in -symmetric quantum mechanics. But this assumption in not
PT
generally true, for unlike the parity operator loses its connection to physical space once
T P
one endows the Hilbert space with an appropriate inner product to reinstate unitarity. This is
because for a general -symmetric Hamiltonian, such as H = p2+x2+ix3, the x-operator is no
PT
longer a physical observable, the kets x do not correspond to localized states in space, and :=
| i P
1
∞ dx x x does not mean space-reflection [2, 3].1 Furthermore, it turns out that one cannot
−∞ | ih− |
actually avoid using the mathematical operations such as Hermitian conjugation (A A†) 2
R
→
or transposition (A At := A† ) in defining the notion of an observable in -symmetric
→ T T PT
quantum mechanics [4, 5].
What makes -symmetry interesting is not its physical appeal but the fact that is an
PT PT
antilinear operator.3 In fact, the spectral properties of -symmetric Hamiltonians [6] that have
PT
made them a focus of recent interest follow from this property. In general, if a linear operator H
commutes with an antilinear operator Θ, the spectrum of H may be shown to be pseudo-real, i.e.,
as a subset of complex plane it is symmetric about the real axis. In particular, nonreal eigenvalues
of H come in complex-conjugate pairs. If H is a diagonalizable operator with a discrete spectrum
the latter condition is necessary and sufficient for the pseudo-Hermiticity of H [7].
In [8], we showed that the spectrum of the Hamiltonian H = p2 + zδ(x) is real and that
one can apply the methods of pseudo-Hermitian quantum mechanics [2] to identify H with the
Hamiltonian of a unitary quantum system provided that the real part of z does not vanish.4 This
Hamiltonian is manifestly non- -symmetric. The purpose of this paper is to offer other classes
PT
of non- -symmetric Hamiltonians that enjoy the same spectral properties.
PT
In the following, we shall use and to denote a (separable) Hilbert space and an invertible
H T
antilinear operator acting in , respectively. For = L2(Rd), we define by [9]
H H T
( ψ)(~x) := ψ(~x)∗, (1)
T
for all ψ L2(Rd) and ~x Rd. For = CN, we identify it with complex-conjugation: For all
∈ ∈ H
~z CN,
∈
~z := ~z∗. (2)
T
2 -Symmetry
QT
Consider a Hamiltonian operator H acting in and commuting with an arbitrary invertible
H
antilinear operator Θ. Because is also invertible and antilinear, we can express Θ as Θ =
T QT
where := Θ is an invertible linear operator. This suggests the investigation of -symmetric
Q T QT
Hamiltonians H,
[H, ] = 0, (3)
QT
1The space reflection operator is given by ∞ dx ξ(x) ξ(−x) where ξ(x) denote the (localized) eigenkets of
−∞ | ih | | i
the pseudo-Hermitian position operator X, [2].
R
2The adjoint A† of an operator A : is defined by the condition ψ Aφ = A† φ where is the
H → H h | i h | i h·|·i
defining inner product of the Hilbert space .
H
3This means that (a ψ +a ψ ) = a∗ ψ +a∗ ψ , where a ,a are complex numbers and ψ ,ψ are
PT 1 1 2 2 1PT 1 2PT 2 1 2 1 2
state vectors.
4Otherwise H has a spectral singularity and it cannot define a unitary time-evolution regardless of the choice
of the inner product.
2
where is an invertible linear operator. Note that need not be a Hermitian operator or an
Q Q
involution, i.e., in general † = and 2 = 1.
Q 6 Q Q 6
We can easily rewrite (3) in the form
H = −1H , (4)
T T Q Q
This is similar to the condition that H is −1-weakly-pseudo-Hermitian [12, 13, 14, 15]:
Q
H† = −1H . (5)
Q Q
In fact, (4) and (5) coincide if and only if
H† = H. (6)
T T
The left-hand side of this relation is the usual “transpose” of H that we denote by Ht. Therefore,
-symmetry is equivalent to −1-weak-pseudo-Hermiticity if and only if Ht = H, i.e., H is
QT Q
symmetric.5 Forexample, let~aandv berespectively complex vector andscalar potentials,~x Rd,
∈
and d Z+. Then the Hamiltonian6
∈
[p~ ~a(~x)]2
H = − +v(~x), (7)
2m
issymmetric ifandonlyif~a =~0. Supposing that~aandv areanalyticfunctions, the -symmetry
QT
of (7), i.e., (4) is equivalent to
[p~+~a(~x)∗]2 [p~ ~a(~x )]2
+v(~x)∗ = Q − Q +v(~x ), (8)
Q
2m 2m
where for any linear operator L : , we have L := −1L . Similarly the −1-weak-
Q
H → H Q Q Q
pseudo-Hermiticity of H, i.e., (5) means
[p~ ~a(~x)∗]2 [p~ ~a(~x )]2
− +v(~x)∗ = Q − Q +v(~x ), (9)
Q
2m 2m
As seen from (8) and (9), there is a one-to-one correspondence between -symmetric and −1-
QT Q
weak-pseudo-Hermitian Hamiltonians of the standard form (7), namely that given such a -
QT
symmetric Hamiltonian H with vector and scalar potentials v and a, there is a −1-weak-pseudo-
Q
Hermitian Hamiltonian H′ with vector and scalar potentials v′ = v and a′ = ia. Note however
that H and H′ are not generally isospectral.
5It is a common practice to identify operators with matrices and define the transpose of an operator H as the
operator whose matrix representation is the transpose of the matrix representation of H. Because one must use a
basis to determine the matrix representation, unlike Ht := H† , this definition of transpose is basis-dependent.
T T
Note however that Ht agrees with this definition if one uses the position basis ~x in L2(Rd) and the standard
{| i}
basis in CN.
6Non-HermitianHamiltoniansofthis formhavebeen usedinmodelling localizationeffects in condensedmatter
physics [16].
3
3 A Class of Matrix Models
Consider two-level matrix models defined on the Hilbert space = C2 endowed with the Eu-
H
clidean inner product . In the following we explore the -symmetry and −1-weak-pseudo-
h·|·i QT Q
a b 1 0
Hermiticity of a general Hamiltonian H = for = , where a,b,c,d,q C.
c d ! Q q 1 ! ∈
3.1 -symmetric Two-Level Systems
QT
Imposing the condition that H is -symmetric (i.e., Eq. (4) holds) restricts q to real and
QT
imaginary values, and leads to the following forms for the Hamiltonian.
For real q:
•
a 0
H = , a,c R. (10)
c a ∈
!
In this case H is a non-diagonalizable operator with a real spectrum consisting of a.
For imaginary q (q = iq with q R 0 ):
• ∈ −{ }
a i bq b
H = − 2 , a,b,c,d R. (11)
c+ i(a d)q d+ i bq ! ∈
2 − 2
In this case the eigenvalues of H are given by E = 1[a + d (a d)2 b(bq2 4c)].
± 2 ± − − −
Therefore, for (a d)2 b(bq2 4c), H is a diagonalizable operator with a real spectrum;
p
− ≥ −
and for (a d)2 < b(bq2 4c), H is diagonalizable but its spectrum consists of a pair of
− −
(complex-conjugate)non-realeigenvalues. Furthermore,thedegeneracycondition: (a d)2 =
−
b(bq2 4c) marks anexceptional spectral point [10, 11]where H becomes non-diagonalizable.
−
In fact, for a = d and b = 0 this condition is satisfied and H takes the form (10). Therefore,
(11) gives the general form of -symmetric Hamiltonians provided that q R.
QT ∈
3.2 1-weakly-pseudo-Hermitian Two-Level Systems
−
Q
Demanding that H is −1-weakly-pseudo-Hermitian does not pose any restriction on the value of
Q
q. It yields the following forms for the Hamiltonian.
For q = 0:
•
a b +ib
H = 1 2 , a,b ,b ,d R. (12)
1 2
b ib d ∈
1 2 !
−
In this case is the identity operator and H = H†. Therefore, H is a diagonalizable
Q
operator with a real spectrum.
4
For q = 0:
• 6
a +ia 2ia2
H = 1 2 − q , a ,a R, q C 0 . (13)
2ia2 a ia 1 2 ∈ ∈ −{ }
q∗ 1 − 2 !
In this case the eigenvalues of H are given by E = a a q −1 4 q 2. Therefore, for
± 1 2
±| || | −| |
q < 2, H is a diagonalizable operator with a real spectrum; and for q > 2, H is diag-
p
| | | |
onalizable but its spectrum consists of a pair of (complex-conjugate) non-real eigenvalues.
Again the degenerate case: q = 2 corresponds to an exceptional point where H becomes
| |
non-diagonalizable.
Comparing(11)with(12)and(13)weseethat -symmetryand −1-weak-pseudo-Hermiticity
QT Q
are totally different conditions on a general non-symmetric Hamiltonian.7 For a symmetric Hamil-
tonian, we can easily show using (12) and (13) that q is either real or imaginary and that H takes
the form (11). The converse is also true, i.e., any symmetric Hamiltonian of the form (11) is either
real (and hence Hermitian) or has the form (13). In summary, -symmetry and −1-weak-
QT Q
pseudo-Hermiticity coincide if and only if the Hamiltonian is a symmetric matrix.
4 Unitary and a Class of non- -Symmetric Hamilto-
Q PT
nians with a Pseudo-Real Spectrum
If is a unitary operator, the −1-weak-pseudo-Hermiticity (5) of a Hamiltonian H implies its
Q Q
-weak-pseudo-Hermiticity, i.e., H† = −1H . This together with (5) leads in turn to
Q Q Q
[H, 2] = 0, (14)
Q
i.e., 2 is a symmetry generator. In the following we examine some simple unitary choices for
Q Q
and determine the form of the −1-weak-pseudo-Hermitian and -symmetric standard Hamil-
Q QT
tonians.
Consider a standard non-Hermitian Hamiltonian (7) in one dimension and let
iℓp
= e ~ (15)
Q
for some ℓ R+. Then introducing
∈
a := (a), a := (a), v := (v), v := (v),
1 2 1 2
ℜ ℑ ℜ ℑ
where and stand for the real and imaginary parts of their argument, and using the identities
ℜ ℑ
−1p = p, −1x = x ℓ, (16)
Q Q Q Q −
7Note that this is not in conflict with the fact that in view of the spectral theorems of [17, 12, 18] both of
these conditionsimply pseudo-Hermiticityofthe Hamiltonianalbeitwithrespectto apseudo-metricoperatorthat
differs from −1, [15].
Q
5
we can express the condition of the −1-weak-pseudo-Hermiticity of H, namely (9), in the form
Q
a (x ℓ) = a (x), a (x ℓ) = a (x), (17)
1 1 2 2
− − −
v (x ℓ) = v (x), v (x ℓ) = v (x). (18)
1 1 2 2
− − −
This means that the real part of the vector and scalar potential are periodic functions with period
ℓ while their imaginary parts are antiperiodic with period ℓ. This confirms (14), for H is invariant
under the translation, x x+2ℓ, generated by 2. We can express a ,v and a ,v in terms of
1 1 2 2
→ Q
their Fourier series. These have respectively the following forms
∞
2nπx 2nπx
ℓ-periodic real parts : c cos +d sin , (19)
1n 1n
ℓ ℓ
n=0(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
∞
(2n+1)πx (2n+1)πx
ℓ-antiperiodic imaginary parts : c cos +d sin , (20)
2n 2n
ℓ ℓ
n=0(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
where c and d are real constants for all k 1,2 and n 0,1,2, .
kn kn
∈ { } ∈ { ···}
Conversely if the real and imaginary parts of both the vector and scalar potential have re-
spectively the form (19) and (20), the Hamiltonian is −1-weak-pseudo-Hermitian. In particular
Q
its spectrum is pseudo-real; its complex eigenvalues come in complex-conjugate pairs. These
Hamiltonians that are generally non- -symmetric acquire -symmetry provided that they
PT QT
are symmetric, i.e., a = a = 0. A simple example is
1 2
p2
H = +λ sin(2kx)+iλ cos(5kx),
1 2
2m
where λ ,λ R and k := ℓ−1 R+.
1 2
∈ ∈
Next, we examine the condition of -symmetry of H, i.e., (8). In view of (16), this condition
QT
is equivalent to (18) and
a (x ℓ) = a (x), a (x ℓ) = a (x), (21)
1 1 2 2
− − −
whichreplaces(17). Thereforev hasthesameformasforthecaseofa −1-weak-pseudo-Hermitian
Q
Hamiltonianbutahasℓ-antiperiodicrealandℓ-periodicimaginaryparts. Inparticular,theFourier
series for real and imaginary parts of a have respectively the form (20) and (19).
We again see that general -symmetric Hamiltonians of the standard form (7) are invariant
QT
under the translation x x+2ℓ. This is indeed to be expected, because in view of [ , ] = 0
→ Q T
we can express (4) in the form
H = −1 H (22)
Q T TQ
and use this identity to establish
2H = H = ( −1 H ) = H 2.
Q QT TQ QT Q T TQ T Q Q
6
The results obtained in this section admit a direct generalization to higher-dimensional stan-
i~ℓ·p~
dard Hamiltonians. This involves identifying with a translation operator of the form e ~ for
Q
some ~ℓ R3 ~0 . It yields -symmetric and −1-weakly-pseudo-Hermitian Hamiltonians with
∈ −{ } QT Q
~
a pseudo-real spectrum that are invariant under the translation ~x ~x 2ℓ.
→ −
An alternative generalization of the results of this section to (two and) three dimensions is to
identify with a rotation operator:
Q iϕnˆ·J~
= e ~ , (23)
Q
where ϕ (0,2π), nˆ is a unit vector in R3, and J~ is the angular momentum operator. Again
∈
[ , ] = 0 and we obtain generally non- -symmetric, −1-weak-pseudo-Hermitian and -
Q T PT Q QT
symmetric Hamiltonians with a pseudo-real spectrum that are invariant under rotations by an
angle 2ϕ about the axis defined by nˆ.
Choosing a cylindrical coordinate system whose z-axis is along nˆ, we can obtain the general
form of such standard Hamiltonians.
The −1-weak-pseudo-Hermiticity of H implies that the real and imaginary parts of the vector
Q
and scalar potentials (that we identify with labels 1 and 2 respectively) satisfy
~a (ρ,θ ϕ,z) =~a (ρ,θ,z), ~a (ρ,θ ϕ,z) = ~a (ρ,θ,z), (24)
1 1 2 2
− − −
v (ρ,θ ϕ,z) = v (ρ,θ,z), v (ρ,θ ϕ,z) = v (ρ,θ,z), (25)
1 1 2 1
− − −
where (ρ,θ,z) stand for cylindrical coordinates. Similarly, the -symmetry yields (25) and
QT
~a (ρ,θ ϕ,z) = ~a (ρ,θ,z), ~a (ρ,θ ϕ,z) =~a (ρ,θ,z). (26)
1 1 2 2
− − −
Again we can derive the general form of the Fourier series for these potentials. Here we suffice to
give the form of the general symmetric Hamiltonian:
p~2 ∞
H = + [e (ρ,z)cos(2nωθ)+f (ρ,z)sin(2nωθ)+
n n
2m
n=0
X
i g (ρ,z)cos[(2n+1)ωθ]+h (ρ,z)sin[(2n+1)ωθ] ], (27)
n n
{ }
where e , f , g , and h are real-valued functions and ω := ϕ−1 R+.
n n n n
∈
5 A -Symmetric and non- -Symmetric Hamiltonian
QT PT
with a Real Spectrum
In the preceding section we examined -symmetric Hamiltonians with a unitary . Because
QT Q
is also a unitary operator, -symmetry with unitary may be considered as a less drastic
P QT Q
generalization of -symmetry. In this section we explore a -symmetric model with a non-
PT QT
unitary .
Q
7
Let a and a† be the bosonic annihilation and creation operators acting in L2(R) and satisfying
[a,a†] = 1, q C 0 , and8
∈ −{ }
:= eqa. (28)
Q
Consider the Hamiltonian operator
H = αa2 +βa†2 +γ a,a† +ma+na†, (29)
{ }
where α,β,γ,m,n C, and demand that H be -symmetric. Inserting (29) in (4) and using
∈ QT
(1) and the identity −1a† = a† q, we obtain
Q Q −
α∗ = α, β∗ = β, γ∗ = γ, m∗ = m 2γq, n∗ = n 2βq, nq = 0.
− −
In particular, because q = 0, we have n = 0 which in turn implies β = 0. Furthermore, assuming
6
that γ = 0, we find that q must be purely imaginary, q = iq with q R 0 , and (m) = γq.
6 ∈ −{ } ℑ
In view of these observations, H takes the following simple form.
H = αa2 +γ a,a† +(µ+iγq)a, (30)
{ }
where µ := (m), α,µ R, and γ,q R 0 .
ℜ ∈ ∈ −{ }
Recalling that a = a and a = a, we see that for µ = 0, the -symmetric Hamil-
P P − T T 6 QT
tonian (30) is non- -symmetric. We also expect that it must have a pseudo-real spectrum. It
PT
turns out that actually the spectrum of H can be computed exactly.
To obtain the spectrum of H we use its representation in the basis consisting of the standard
normalized eigenvectors n ofthe number operatora†a. Using thefollowing well-known properties
| i
of n , [19],
| i
a n = √n n 1 , a† n = √n+1 n+1 ,
| i | − i | i | i
we find for all m,n 0,1,2, ,
∈ { ···}
H := m H n = γ(2n+1)δ +(µ+iγq)√nδ +α n(n 1)δ .
mn mn m,n−1 m,n−2
h | | i −
p
As seen from this relation the matrix (H ) is upper-triangular with distinct diagonal entries and
mn
up to three nonzero terms in each row. This implies that the eigenvalues E of (H ) are identical
n mn
with its diagonal entries, i.e., E = γ(2n + 1). In particular, H has a discrete, equally spaced,
n
real spectrum that is positive for γ > 0. In the latter case H is isospectral to a simple harmonic
oscillator Hamiltonian with ground state energy γ.
It is not difficult to see that for each n 0,1,2, the span of 0 , 1 , , n , which
∈ { ···} {| i | i ··· | i}
we denote by , is an invariant subspace of H. This observation allows for the construction of
n
H
a complete set of eigenvectors of H and establishes the fact that H is a diagonalizable operator
with a discrete real spectrum. Therefore, in view of a theorem proven in [20], it is related to a
Hermitian operator via a similarity transformation, i.e., it is quasi-Hermitian [21].
8The considered in Section 3 may be viewed as a fermionic analog of (28).
Q
8
The existence of the invariant subspace also implies that the eigenvectors ψ of H corre-
n n
H | i
sponding to the eigenvalue E belong to , i.e., ψ is a linear combination of 0 , 1 , , n 1
n n n
H | i | i | i ··· | − i
and n . This in turn allows for a calculation of ψ . For example,
n
| i | i
2γ
ψ = c 0 , ψ = c 0 + 1 ,
0 0 1 1
| i | i | i | i m | i
(cid:20) (cid:18) (cid:19) (cid:21)
4γm 2√2γ2
ψ = c 0 + 1 + 2 ,
| 2i 2 | i m2 +αγ | i m2 +αγ | i
" ! #
(cid:18) (cid:19)
where c ,c ,c are arbitrary nonzero normalization constants and m = µ+iγq.
0 1 2
6 Concluding Remarks
Itisoftenstatedthat -symmetryisaspecialcaseofpseudo-Hermiticity because -symmetric
PT PT
Hamiltonians are manifestly -pseudo-Hermitian. This reasoning is only valid for symmetric
P
Hamiltonians H that satisfy H† = H . In general to establish the claim that -symmetry
T T PT
is a special case of pseudo-Hermiticity one needs to make use of the spectral theorems of [17,
12, 18]. Indeed what makes -symmetric Hamiltonians interesting is the pseudo-reality of their
PT
spectrum. This is a general property of all Hamiltonians that are weakly pseudo-Hermitian or
possess a symmetry that is generated by an invertible antilinear operator. We call the latter
-symmetric.
QT
In this article, we have examined in some detail the similarities and differences between -
QT
symmetry and −1-weak-pseudo-Hermiticity and obtained large classes of symmetric as well
Q
as asymmetric non- -symmetric Hamiltonians that share the spectral properties of the -
PT PT
symmetric Hamiltonians. In particular, we considered the case that is a unitary operator and
Q
showed that in this case -symmetry and −1-weak-pseudo-Hermiticity imply 2-symmetry
QT Q Q
of the Hamiltonian. We also explored a concrete example of a -symmetric Hamiltonian with
QT
a non-unitary that is not -symmetric. We determined the spectrum of this Hamiltonian,
Q PT
established its diagonalizability, and showed that it is indeed quasi-Hermitian.
9
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