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A foundational approach to the meaning of time reversal
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Abstract Wigner [12] introduced a ‘foundational’ account
to the meaning of time reversal, by assuming quantum mechanics is time reversal invariant, and then arguing that T is
antiunitary. This account was incomplete, because it used a
different strategy to characterize the particular transformation properties of an observable. In this paper, we extend
Wigner’s account, by using it to derive the transformation
rules for the canonical and angular momentum observables.
We then show how the same approach can be used to derive the time reversal operator in classical mechanics and
Bohmian mechanics as well.
Wigner [12] and later R. G. Sachs [8] advocated a different approach to antiunitarity, arguing that it is a consequence of the time reversibility of the quantum mechanics of free motion. However, both continued to assume the
transformation rules above without justification. We would
like to extend the Wigner-Sachs approach, by using it to derive the transformation rules for observables on the basis of
their commutation relations. We then show that the same approach can be used to derive the time reversal operator in
classical mechanics and in Bohmian mechanics.
Keywords quantum mechanics · symmetry · time reversal
2 A foundational approach
1 Introduction
We begin with a minimal necessary condition on time reversal, which requires that time-orderings be reversed, but
leaves open the exact way in which T transforms a state.
Schwinger [9, p.151] demanded that in building up quantum
theory, “classical mechanics terminology... is all that we borrow”. From this perspective, the treatment of time reversal in
quantum mechanics by way of its classical analogue is not
satisfactory. This treatment is adopted in many introductory
textbooks (e.g. [1], [5], [6], [8] and [10]). It begins by assuming that time reversal transforms observables like their
classical analogues:
X 7→ X
P 7→ −P
J 7→ −J,
(1)
where the latter follows from the former two. One then proceeds to derive the antiunitarity of time reversal from the
commutation relations. Mathematically, this approach fails
to be rigorous for observables that have no classical analogue. Philosophically, it fails to provide us with a foundational approach to the meaning of time reversal, given that
quantum mechanics is the more fundamental theory.
Address(es) of author(s) should be given
Definition 1 Given a dynamical evolution through state space1
Γ = {ψ } along a single parameter t, it is necessary (though
not sufficient) that the time reversal mapping send each trajectory ψ (t) to T ψ (−t), where T is a bijection on Γ .
The symmetry principle that we adopt is the following2 .
Claim (Free Motion Symmetry) In the absence of forces and
interactions, the isometries of the background spacetime must
be norm-preserving symmetries of the equations of motion.
In particular, we demand all free particles and free fields be
invariant under time reversal, while leaving open whether or
not interacting matter is time reversible. Note that by time
reversibility we mean:
Definition 2 A dynamical theory is T reversal invariant if,
whenever ψ (t) is a solution, so is T ψ (−t), where T is normpreserving3 .
1 By ‘state space’ we mean the general space of state that a dynamical system passes through over time. In classical mechanics this it is
generally phase space, and in quantum mechanics it is Hilbert space.
2 A similar principle has been called “kinematic admissibility” by
Sachs [8] and “microreversibility” by Messiah [6].
3 In theories in which there is no natural norm, the latter requirement
may be discarded.
2
Wigner argued that time reversal is antiunitary on the basis of this kind of assumption, without assuming the transformation rules (1). In the following, we formulate and prove
a precise version of Wigner’s claim.
Proposition 1 For the free particle Hamiltonian H0 , suppose that Schrödinger evolution is invariant under the normpreserving bijection T that takes ψ (t) 7→ T ψ (−t) (Free Motion Symmetry). Suppose further that the spectrum of H0 is
positive. Then T is antiunitary.
Proof Let ψ (t) = e−itH0 ψ0 describe a dynamical trajectory,
with initial state psi0 and free Hamiltonian H0 . Substituting
t 7→ −t, we derive an equivalent formulation:
ψ (−t) = eitH0 ψ0 .
(2)
Now, the assumption of Free Motion Symmetry says the
time-reversed trajectory T ψ (−t) with initial state T ψ0 also
satisfies normal Schrödinger evolution:
T ψ (−t) = e−itH0 T ψ0 .
(3)
But substituting (2) into the LHS of (3) gives:
TeitH0 ψ0 = e−itH0 T ψ0 .
Tayor expanding the exponentials, we see that
In particular:
|hψ , X φ i| = |hT ψ , T X φ i|
= ψ , T †T X φ = ψ , (T † T X)† φ = ψ , (T X)† T φ = |hψ , T XT φ i|
= ψ , T XT −1 φ (T is antiunitary)
(by Equation (4))
(by Equation (4))
(T 2 = eiα I).
By Theorem II of [6, §XV.2], this implies that TAT −1 =
eiθ A. Now, let X and P be any observables such that [X, P] =
i. By the above, we may write T XT −1 = eiθx X and T PT −1 =
eiθ p P. But since T is antilinear, we can use the fact that
TiT −1 = −i to get:
−(XP − PX) = −i
= TiT −1
= T (XP − PX)T −1
= (T XT −1 )(T PT −1 ) − (T PT −1 )(T XT −1 )
= eiθx eiθ p (XP − PX).
Thus eiθx eiθ p = −1, and so eiθ p = −e−iθx . Finally, since the
spectrum of X is real, we have that eiθx = ±1, and the conclusion follows. ⊓
⊔
1 2
1
t T H02 +· · · ) = (T −iH0tT − t 2 H02 T +· · · ), The foundational approach thus allows us to derive both time
2!
2!
reversal operators in ordinary quantum mechanics. The first
and in particular that TiH0 = −iH0 T . But T satisfies the con- is the standard T operator. The second is the standard PT opditions of Wigner’s theorem, and thus is either unitary or an- erator. An analogous argument allows us to derive the transtiunitary by Wigner’s theorem. Moreover, T cannot be uni- formation properties of angular momentum.
tary. For if it were, then we would have that TiH0 = iT H0 =
of Proposition 5,
−iH0 T , and hence that T H0 = −H0 T . But this implies that Proposition 3 In addition to the premises
2 = eiθ I), and S , S and
suppose
that
T
is
an
involution
(T
x y
for any eigenstate ψ of H0 with positive energy e, the transS
are
Hermitian
operators
satisfying
the
angular
momen−1
z
formed state T ψ would be an eigenstate of T H0 T with
−1 = −S , T S T −1 =
tum
commutation
relations.
Then
T
S
T
x
x
y
negative energy −e, contradicting our assumption. There−Sy , and T Sz T −1 = −Sz .
fore, T must be antiunitary. ⊓
⊔
(T +TiH0t −
Wigner fell short of applying his account to characterize the particular transformation rules of an observable. We
complete Wigner’s argument, first by extending it to derive
the transformation properties of observables satisfying the
canonical commutation relations.
Proposition 2 In addition to the premises of Proposition 5,
suppose that T is an involution (T 2 = eiθ I), and X, P are
Hermitian operators satisfying [X, P] = i. Then T XT −1 =
±X and T PT −1 = ∓P.
Proof Let A be an arbitrary observable. Proposition 1 establishes that T is antiunitary, and hence antilinear. We first
observe that if T is an involution, then for every A there is
some θ such that TAT −1 = eiθ A. This makes use of the fact
that for any A and for all ψ , φ ,
∗
hψ , Aφ i = ψ , A† φ .
(4)
Proof Since T is an involution, we may apply the argument
of Proposition 2 and conclude that for any observable A,
there is a real angle θ such that TAT −1 = eiθ A. The canonical anticommutation relations now state that:
Sx2 = Sy2 = Sz2 = 1
(5)
2iεxyz Sz = [Sx , Sy ] = Sx Sy − Sy Sx
(6)
where εxyz is +1 for even permutations and −1 otherwise.
Equation (5) implies that, for each α = x, y, z,
T Sα2 T −1 = (T Sα T −1 )2 = e2iθα Sα2 = 1,
and hence that eiθα = ±1. We can now apply Equation (6)
to see that eiθα = −1 is the only option. Using the fact that
T 2iSz T −1 = −2iT Sz T −1 , we have:
−2iT Sz T −1 = T [Sx , Sy ]T −1 = [T Sx T −1 , T Sy T −1 ].
3
But we may write T Sα T −1 = eiθα for each α = x, y, z, and
thus,
But by applying the same argument to the other even permutations of xyz in Equation (6) implies
Proof Let x = |xi be a position eigenvector. We begin by
observing that although the Bohmian time reversal operator
T acts on position values in configuration space, we have
further assumed that time reversal acts invariantly on the set
of position eigenvectors, T {x} = {x}. We will use the same
notation T to represent the two different operators, keeping
in mind that one acts on Hilbert space while the other acts
on configuration space.
The assumption of Free Motion Symmetry implies that
the time reversed values of x and ψ (x) satisfy the Bohmian
guidance equation:
θx − θy + θz = π
−θx + θy + θz = π .
dT x
1 (T ∇) hT x, T ψ i
.
= − Im
hT x, T ψ i
dt
m
eiθx eiθy [Sx , Sy ] = −eiθz 2iSz
= −eiθz [Sx , Sy ].
Hence eiθx eiθy = −eiθz , which is only satisfied if
θx + θy − θz = π .
(8)
Summing these equations shows that θx = θy = θz = π . There- We will first show how this equation can be simplified considerably. Our conclusion will then follow immediately from
⊔
fore, T Sα T −1 = −Sα , for each α = x, y, z. ⊓
the assumption that T is an involution.
It is of some interest that this proposition provides a prinThe wavefunction can only be reversed in one of two
cipled derivation of the transformation rules for the spin ob- ways: either hT x, T ψ i = hx, ψ i (if T is linear), or hT x, T ψ i =
servable in ordinary quantum mechanics. The argument for hx, ψ i∗ (if T is antilinear). If T is linear, then we can expand
the standard transformation rules (1) does not, since the spin ψ in the position basis to get:
degree of freedom does not have a classical analogue.
Z ′ ′
′
hT x, T ψ i = T x,
x , ψ T x dx
Z 3 Extension to other theories
=
x′ , ψ T x, T x′ dx′
There exist quantum theories for which Wigner’s approach
Z is not applicable. Recall that our Definition 1 demanded dy=
x′ , ψ δ (T x − T x′ )dx′ ,
namical evolution be characterized by a single evolution parameter t. As a consequence, this account does not say any- where the last line follows from the assumption that T acts
thing about theories like GRW [4], CSL [7], or any other invariantly on the set of position eigenvectors, preserving
theory that incorporates an additional stochastic parameter their orthonormality. We thus have that hT x, T ψ i = hx, ψ i.
into the dynamics.
It follows similarly that hT x, T ψ i = hx, ψ i∗ when T is antiHowever, the Wigner’s approach can be applied to the linear.
approach quantum mechanics developed by Bohm4 , and it
Next, we observe that by the chain rule,
leads to results analogous to the ones we have seen above.
∂
∂x ∂
∂x
One difference is that the Bohmian time reversal operator
T∇ =
=
=
∇.
acts on the positions of Bohmian particles, while the guid∂Tx ∂Tx ∂x ∂Tx
ance equation
Combing these two results, we now find that Equation (8)
dx
1 ∇ψ (x)
= − Im
.
(7) reduces to:
dt
m
ψ (x)
dx
1 ∇ hx, ψ i
dT x
depends on a Hilbert space wavefunction. To deal with this
,
=±
Im
hx, ψ i
dt
dT x m
subtlety, we must adopt the minimal assumption that position basis-vectors in H be considered as functions of posi- where we get a ‘+’ if T is antilinear and a ‘−’ if T is linear.
tions in configuration space. We then take the time reversal But the RHS is just that of the usual guidance equation, with
operator to act invariantly on this set of positions: T |xi = an extra factor ∓(dx/dT x). Therefore, we can substitute in
|x′ i. For simplicity, we illustrate for the case of a single pardx
1 ∇ hx, ψ i
ticle on a string.
= − Im
,
hx, ψ i
dt
m
Proposition 4 Suppose that the Bohmian guidance equation is time reversal invariant for the free particle Hamiltonian (Free Motion Symmetry), and that T is an involution on
configuration space, which acts invariantly on set of position
eigenvectors, T {|xi} = {|xi}. Then there are only two time
reversal operators: either T x = x and T (dx/dt) = −dx/dt,
or else T x = x0 − x and T (dx/dt) = dx/dt for some fixed x0 .
4
See [3] for an introduction.
to get that:
dx dx
dT x
=∓
.
dt
dT x dt
Multiplying by inverses, this implies,
dT x dt dT x
dT x 2
=
= ∓1.
dt dx dx
dx
4
But T is real-valued, so the −1 case is impossible. This establishes that T must be an antilinear Hilbert space operator.
Finally, since dT x/dx = ±1, we find by integration that
determine how T operates on q, we substitute T p(−t) =
cp(−t) into (11):
d
1
T q(−t) = cp(−t)
dt
m
d
= −c q(−t)
dt
T x = x0 ± x
for some constant x0 . But since we have assumed that T is
involution, this reduces to only two options: either T x = x, or
else T x = x0 − x. It is now easily seen that our definition of
time reversal transforms dx/dt identically in the latter case,
and to its negative in the former case (because time reversal
sends t 7→ −t). ⊓
⊔
Note that our approach relies on there being some reason to accept the Bohmian guidance equation over the many
other possible guidance equations5 . Some [2] have proposed
that the guidance equation be derived from the demand of
Galilei invariance. However, this technique presupposes a
particular meaning for the time reversal operator, which from
our perspective is unjustified. On a foundational approach
to time reversal, Bohmians must rely on some independent
means of establishing the correctness of the guidance equation.
We conclude by noting that the foundational approach
can also be used to derive the two time reversal operators in
classical Hamiltonian mechanics: the standard T operator,
or the standard PT operator. This suggests that the principle
of Free Motion Symmetry has some degree of applicability
outside of quantum mechanics as well.
Proposition 5 For the free particle Hamiltonian, suppose
that Hamilton’s equations are invariant under a linear bijection T that takes x(t) 7→ T x(−t) and p(t) 7→ T p(−t) (Free
Motion Symmetry), where T is a linear involution. Then either T p = −p and T q = q, or else T p = p and T q = q0 − q
for some fixed q0 .
where we have substituted (9) in the last line. Integrating,
we see that T q(−t) = −cq(−t) + q0 for some constant q0 .
Therefore, T p = cp and T q = q0 −cq. To complete the proof,
we now assume that T is a linear involution. Then:
p = T 2 p = T (cp) = c2 p,
so c2 = 1. Moreover,
q = T 2 q = T (q0 − cq)
= q0 (1 − c) + q.
(13)
The constant term q0 (1 − c) must therefore vanish. Since
c2 = 1 and c is real, there are two options. If c = −1, then
q0 = 0. Then T p = −p and T q = q. On the other hand, if
c = 1, then any real q0 will satisfy (13). Then we get a class
of time-reversal operators indexed by q0 , namely, T p = p
and T q = q0 − q. ⊓
⊔
Note that the constant q0 just represents our freedom to choose
the axis about which we define parity reversal.
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d
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(9)
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(10)
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1
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T q(−t) = T p(−t)
(11)
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(12) 10. Ramamurti Shankar. Principles of Quantum Mechanics. New
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We now note that (10) and (12) can be integrated to get
T p(−t) = cp(−t). This tells us how T operates on p. To 12. Eugene Wigner. Group Theory and its Application to the Quantum
Mechanics of Atomic Spectra. New York: Academic Press (1959),
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5 See [11] for an overview.