18
2 Function Spaces and Matrices
This equation decomposes the closed path in time in four consecutive steps. Reading
Eq. (2.29) from right to left, one sets off at time t 0 and reverses time ( ˆ
ϑ −1 ). Now
time runs for a certain interval t along the reversed time axis. A positive interval
actually means that we are returning in time since the time axis has been oriented
toward the past. This operation is presented by the displacement ˆ
T t . Then one
applies the time reversal again and now runs forward over the same interval to close
the loop. The forward translation corresponds to the same ˆ
T t operator since again
the interval is positive. Now multiply both sides of the equation, on the right, by
ˆ
ϑ ˆ
T −t :
ˆ
T t ˆ
ϑ = ˆ
ϑ ˆ
T −t
(2.30)
The actions of the translations on the wavefunction are given by
T t Ψ(t 0 ) = Ψ(t 0 ) exp
−
iEEt
T −t Ψ(t 0 ) = Ψ(t 0 ) exp
iEEt
(2.31)
Applying now both sides of Eq. (2.30) to the initial state yields
T t ϑΨ(t 0 ) = ϑT −t Ψ(t 0 )
= ϑ exp
iEEt
Ψ(t 0 )
(2.32)
Since the Hamiltonian that we have used is invariant under time reversal, the function ϑΨ(t 0 ) on the left-hand side of Eq. (2.32) will be characterized by the same
energy, E, and thus translate in time with the same phase factor as Ψ(t 0 ) itself.
Then the equation becomes
exp
−
iEEt
ϑΨ(t 0 ) = ϑ exp
iEEt
Ψ(t 0 )
(2.33)
which shows that time reversal will invert scalar constants to their complex conjugate, and hence it will be an anti-linear operator.
Note that in the present derivation we avoided providing an explicit form for the
inverse of the time reversal operator. As a matter of fact, while space inversion is its
own inverse, applying time reversal twice may give rise to an additional phase factor,
which is +1 for systems with an even number of electrons, but −1 for systems with
an odd number of electrons. We shall demonstrate this point later in Sect. 7.6. Hence,
ϑ −1 =±ϑ ,or
ˆ
ϑ
2 =±1
(2.34)
2 Function Spaces and Matrices
This equation decomposes the closed path in time in four consecutive steps. Reading
Eq. (2.29) from right to left, one sets off at time t 0 and reverses time ( ˆ
ϑ −1 ). Now
time runs for a certain interval t along the reversed time axis. A positive interval
actually means that we are returning in time since the time axis has been oriented
toward the past. This operation is presented by the displacement ˆ
T t . Then one
applies the time reversal again and now runs forward over the same interval to close
the loop. The forward translation corresponds to the same ˆ
T t operator since again
the interval is positive. Now multiply both sides of the equation, on the right, by
ˆ
ϑ ˆ
T −t :
ˆ
T t ˆ
ϑ = ˆ
ϑ ˆ
T −t
(2.30)
The actions of the translations on the wavefunction are given by
T t Ψ(t 0 ) = Ψ(t 0 ) exp
−
iEEt
T −t Ψ(t 0 ) = Ψ(t 0 ) exp
iEEt
(2.31)
Applying now both sides of Eq. (2.30) to the initial state yields
T t ϑΨ(t 0 ) = ϑT −t Ψ(t 0 )
= ϑ exp
iEEt
Ψ(t 0 )
(2.32)
Since the Hamiltonian that we have used is invariant under time reversal, the function ϑΨ(t 0 ) on the left-hand side of Eq. (2.32) will be characterized by the same
energy, E, and thus translate in time with the same phase factor as Ψ(t 0 ) itself.
Then the equation becomes
exp
−
iEEt
ϑΨ(t 0 ) = ϑ exp
iEEt
Ψ(t 0 )
(2.33)
which shows that time reversal will invert scalar constants to their complex conjugate, and hence it will be an anti-linear operator.
Note that in the present derivation we avoided providing an explicit form for the
inverse of the time reversal operator. As a matter of fact, while space inversion is its
own inverse, applying time reversal twice may give rise to an additional phase factor,
which is +1 for systems with an even number of electrons, but −1 for systems with
an odd number of electrons. We shall demonstrate this point later in Sect. 7.6. Hence,
ϑ −1 =±ϑ ,or
ˆ
ϑ
2 =±1
(2.34)