Understanding Quantum Physics: An Advanced Guide for the Perplexed

the
active and passive pictures is due to the fact that moving the system one way
is equivalent to moving the coordinate system the other way by an equal amount
(see also Shankar 1994). In the following, we will mainly analyze spacetime
translations in terms of active transformations.
    A space
translation operator can be defined as
    T (a)ψ(x, t) = ψ(x
− a, t). (3.1)
    It means
translating rigidly the state of a system, ψ(x, t), by an amount a in the
positive x direction. The operator preserves the norm of the state because ∫ψ ∗ (x, t)ψ(x, t)dx = ∫ψ ∗ (x − a, t)ψ(x − a, t)dx. This implies that
T (a) is unitary, satisfying T † (a)T (a) = I. As a unitary operator,
T (a) can be further expressed as
    T (a) = e −iaP ,
(3.2) where P is called the generator of space translation, and it is Hermitian
and its eigenvalues are real. By expanding ψ(x − a, t) in order of a, we can
further get
    P = −i∂/∂x. (3.3)
    Similarly, a time
translation operator can be defined as
    U(t)ψ(x, 0) = ψ(x,
t). (3.4)
    Let the evolution
equation of state be of the following form:
    i∂ψ(x, t)/∂t =
Hψ(x, t). (3.5)
    where H is a
to-be-determined operator that depends on the properties of the system. In the
following analysis of this section, we assume H is independent of the evolved
state, namely the evolution is linear [35] . Then the time translation operator U(t)
can be expressed as U(t) = e −itH , and H is the generator of time
translation. Note that we cannot determine whether U(t) is unitary and H is
Hermitian here.
    Let’s now analyze
the implications of spacetime translation invariance for the law of motion of a
free system or an isolated system. First, time translational invariance
requires that H has no time dependence, namely dH/dt = 0. This can be
demonstrated as follows (see also Shankar 1994, p.295). Suppose an isolated
system is in state ψ 0 at time t 1 and evolves for an
infinitesimal time δt. The state of the system at time t 1 + δt, to
first order in δt, will be
    ψ(x, t 1 + δt) = [I − iδtH(t 1 )]ψ 0 (3.6)
    If the evolution
is repeated at time t 2 , beginning with the same initial state, the
state at t 2 + δt will be
    ψ(x, t 2 + δt) = [I − iδtH(t 2 )]ψ 0 (3.7)
    Time translational
invariance requires the outcome state should be the same:
    ψ (x, t 2 + δt) − ψ(x, t 1 + δt) = iδt[H(t 1 ) − H(t 2 )]ψ 0 = 0 (3.8)
    Since the initial
state ψ 0 is arbitrary, it follows that H(t 1 ) = H(t 2 ).
Moreover, since t 1 and t 2 are also arbitrary, it follows
that H is time-independent, namely dH/dt = 0. It can be seen that this result
relies on the linearity of evolution. If H depends on the state, then obviously
we cannot obtain dH/dt = 0 because the state is related to time, though we
still have H(t 1 , ψ 0 ) = H(t 2 , ψ 0 ),
which means that the state-dependent H also satisfies time translational
invariance.
    Secondly, space
translational invariance requires [ T (a), U(t)] = 0, which further leads to
[P, H] = 0. This can be demonstrated as follows (see also Shankar 1994, p.293).
Suppose at t = 0 two observers A and B prepare identical isolated systems at x
= 0 and x = a, respectively. Let ψ(x, 0) be the state of the system prepared by
A. Then T (a)ψ(x, 0) is the state of the system prepared by B, which is obtained
by translating (without distortion) the state ψ(x, 0) by an amount a to the
right. The two systems look identical to the observers who prepared them. After
time t, the states evolve into U(t)ψ(x, 0) and U(t)T (a)ψ(x, 0). Since the time
evolution of each identical system at different places should appear the same
to the local observers, the above two systems, which differed only by a spatial
translation at t = 0, should differ only by the same spatial translation at
future times. Thus the state U(t)T (a)ψ(x, 0) should be the translated version
of A’s system at time t, namely we have U(t)T (a)ψ(x, 0) = T (a)U(t)ψ(x, 0).
This relation holds true for any initial state ψ(x, 0), and thus we have [T
(a), U(t)] = 0, which says that space translation operator and time translation
operator are commutative. Again, we stress that the linearity of evolution is
an important presupposition of this result. If U(t) depends on the state, then
the space translational invariance will only lead to U(t, T ψ)T (a)ψ(x, 0) = T
(a)U(t, ψ)ψ(x, 0), from which we cannot obtain [T (a), U(t)] = 0.
    When dH/dt = 0,
the solutions of the evolution equation Eq.(3.5) assume the following