To:
P. Rungta
From:
C. M. Caves
Subject:
Antilinear operators
2001 May 6
The following notes are an elaboration of Chap. XV.I of Messiah and of CMC notes
dated 00–7–12.
I. General considerations
An
antilinear operator
K
:

ψ
i → K
ψ
i
acts on linear combinations according to
K
(
α
1

ψ
1
i
+
α
2

ψ
2
i
)
=
α
*
1
(
K
ψ
1
i
)
+
α
*
2
(
K
ψ
2
i
)
.
(1)
A product of linear and antilinear operators is linear if it has an even number of antilinears
and antilinear if it has an odd number of antilinears.
The left action of an antilinear operator, i.e.,
h
φ
 → h
φ
K
, is given by
(
h
φ
K
)

ψ
i
=
£
h
φ

(
K
ψ
i
)/
*
.
(2)
The complex conjugation is present so that
£(
h
φ
1

α
*
1
+
h
φ
2

α
*
2
)
K
/

ψ
i
=
£(
h
φ
1

α
*
1
+
h
φ
2

α
*
2
)(
K
ψ
i
)/
*
=
α
1
£
h
φ
1

(
K
ψ
i
)/
*
+
α
2
£
h
φ
2

(
K
ψ
i
)/
*
=
α
1
(
h
φ
1
K
)

ψ
i
+
α
2
(
h
φ
2
K
)

ψ
i
=
£(
h
φ
1
K
)
α
1
/

ψ
i
+
£(
h
φ
2
K
)
α
2
/

ψ
i
=
£(
h
φ
1
K
)
α
1
+
(
h
φ
2
K
)
α
2
]

ψ
i
,
(3)
i.e., so that
K
is antilinear to the left:
(
h
φ
1

α
*
1
+
h
φ
2

α
*
2
)
K
=
(
h
φ
1
K
)
α
1
+
(
h
φ
2
K
)
α
2
.
(4)
The presence of the complex conjugation in Eq. (2) means that in a matrix element one
must always indicate explicitly whether an antilinear operator acts to the right or to the
left.
The adjoint (Hermitian conjugate) of an antilinear operator is defined in the same
way as for a linear operator, i.e.,
(
K
ψ
i
)
†
=
(
h
ψ
K
†
)
,
(5)
but this means that
h
φ

(
K
ψ
i
)
=
£(
h
ψ
K
†
)

φ
i
/
*
=
h
ψ

(
K
†

φ
i
)
.
(6)
1
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This definition of adjoint means that the adjoint works the same way on products as
for linear operators. To verify this, we need to check two cases, keeping in mind that the
product of two antilinear operators is a linear operator and the product of an antilinear
operator and a linear is antilinear:
h
ψ

(
KL
)
†

φ
i
=
h
φ

(
KL
)

ψ
i
*
= (
h
φ
K
)(
L
ψ
i
) = [(
h
ψ
L
†
)(
K
†

φ
i
)]
*
=
h
ψ

(
L
†
K
†

φ
i
)
,
h
ψ

[(
K
A
)
†

φ
i
] =
h
φ

(
K
A

ψ
i
) = [(
h
φ
K
)(
A

ψ
i
)]
*
= (
h
ψ

A
†
)(
K
†

φ
i
) =
h
ψ

(
A
†
K
†

φ
i
)
.
(7)
Thus we have that (
KL
)
†
=
L
†
K
†
and (
K
A
)
†
=
A
†
K
†
. Notice that the linear operators
K
†
K
and
KK
†
are Hermitian and, indeed, positive.
An antilinear operator is specified by its “matrix elements” in any orthonormal basis

e
j
i
:
h
e
j

(
K
e
k
i
)
=
£(
h
e
j
K
)

e
k
i
/
*
=
h
e
k

(
K
†

e
j
i
)
.
(8)
Given an orthonormal basis, there is an associated linear operator
A
that acts the same
way as
K
in this basis,
A

e
j
i
=
K
e
j
i
,
(9)
and thus has the same matrix elements in this basis:
h
e
j

A

e
k
i
=
h
e
j

(
K
e
k
i
)
.
(10)
The relation between
K
and
A
becomes clear when
K
acts on an arbitrary vector
K
ψ
i
=
K
ˆ
X
j
c
j

e
j
i
!
=
X
j
c
*
j
K
e
j
i
=
X
j
c
*
j
A

e
j
i
=
A
ˆ
X
j
c
*
j

e
j
i
!
=
A

ψ
*
i
=
A
C
ψ
i
=
C
A
*

ψ
i
.
(11)
Here the complex conjugations on

ψ
i
and
A
stand for complex conjugation in the chosen
basis

e
j
i
, and
C
is the antilinear operator that complex conjugates in this basis.
The
antilinear operator can be written simply as
K
=
A
C
=
C
A
*
.
(12)
The matrix elements of
C
are
δ
jk
=
h
e
j

(
C
e
k
i
)
=
£(
h
e
j
C
)

e
k
i
/
*
=
h
e
k

(
C
†

e
j
i
)
.
(13)
We can conclude that
(
h
e
j
C
)
=
h
e
j

,
(14)
which means that when acting to the left,
C
complex conjugates in the same basis. We
can also conclude that
C
†

e
j
i
=

e
j
i
,
(15)
2
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 Spring '09
 Linear Algebra, basis, Orthogonal matrix, Hilbert space, λj λj, Vjk

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