The simple failure of Curie’s principle

Bryan W. Roberts
usc.edu/bryanroberts

Forthcoming in Philosophy of Science

July 3, 2013

Abstract. I point out a simple sense in which the standard for-
mulation of Curie’s principle is false, when the symmetry transfor-
mation it describes is time reversal.

1. Introduction

John Earman has suggested that there is a simple formulation of
Curie’s principle that is not only deeply intuitive but “virtually ana-
lytic” (Earman 2004, p.173). He is not the only one to take this view1,
but gives one of its clearest statements. Earman formulates Curie’s
principle as the claim: If,

(CP1) the laws of motion/field equations governing the system are
deterministic;

(CP2) the laws of motion/field equations governing the system are
invariant under a symmetry transformation; and

Acknowledgements. For their helpful comments and suggestions I would like to
thank John Earman, John D. Norton, and the December 8, 2012 audience at the
Southern California Philosophy of Physics Reading Group at U.C., Irvine.

1Curie himself took it to be an a priori truth that, “[w]hen certain effects show
a certain asymmetry, this asymmetry must be found in the causes which gave
rise to them” (Curie 1894), translation from (Brading and Castellani 2003, p.311-
313). The meaning of this statement is vague, because the words “cause” and
“effect” here are not defined. But on the plausible reading that an “effect” is a
state, and a “cause” is another state related to the first by a deterministic law,
we immediately get the formulation presented by Earman. Similar readings of
Curie can be found in Mittelstaedt and Weingartner (2005, p.231), where it is said
that, “from an asymmetric effect and symmetric laws we may conclude asymmetric
initial conditions,” and Ismael (1997, p.170), who claims to have proven that “all
characteristic symmetries of a Curie-cause are also characteristic symmetries of its
effect.”

1

http://www.usc.edu/bryanroberts


2 Bryan W. Roberts

(CP3) the initial state of the system is invariant under said symmetry;
then

(CP4) the final state of the system is also invariant under said sym-
metry. (Earman 2004, p.176)

Speaking intuitively, one might summarize the principle: if no asym-
metry goes in, then no asymmetry comes out.

I would like to point out a simple sense in which this formulation
of Curie’s principle fails, when the language therein is interpreted in
the standard way, and the symmetry transformation is time reversal.
I will begin by illustrating a very simple counterexample in classical
Hamiltonian mechanics, and then show how such counterexamples are
endemic to quantum mechanics and quantum field theory. I discuss
three alternative interpretations of Curie’s principle that aim to resist
the conclusion that the principle fails for time reversal, and argue that
none are satisfying. I conclude that one must apply Curie’s principle
with care, as it only applies for a particular kind of symmetry trans-
formation that does not include time reversal.

2. Failure in classical mechanics

2.1. In pictures. Take a harmonic oscillator, like a bob on a spring. It
is manifestly time reversal invariant, in that for every possible motion
of the bob, there is a “time-reversed motion” that is also possible.

How does the instantaneous state of an ordinary classical system like
this one change under time reversal? The standard textbook answer is
that the position of the state remains unchanged, while the direction
of the momentum is reversed. One can get the intuition for this by
imagining that we film the motion of the bob, and then play the film
in reverse. To determine what time reversal does at an instant we look
at a single “frame” of the film and ask how it changes when we view
the reversed film. The answer is that, since rightward motion in the
original film becomes leftward in the reversed film, the momenta simply
reverse sign. We thus say that an instantaneous state of an ordinary
classical system is “invariant” (or “unchanged” or “preserved”) under
time reversal if and only if the momentum of that state is zero.

Let us now suppose that this particular bob-spring system begins its
motion at time t = 0 with the spring compressed out of equilibrium,
and with no initial momentum, as in Figure 1(a). The bob then springs
back in the other direction, acquiring some non-zero momentum, as in
Figure 1(b). How does time reversal transform these initial and final
states?



The simple failure of Curie’s principle 3

(a) Initial state (b) Final state

Figure 1. (a) A harmonic oscillator initially compressed out
of equilibrium with zero momentum. (b) A final state for
which the system has non-zero momentum.

(a) Time reversed initial
state

(b) Time reversed final state

Figure 2. (a) The initial state has no momentum, and so
is preserved by time reversal. (b) A final state has non-zero
momentum, and so is not preserved by time reversal.

Our initial state has zero momentum, so it is preserved by the time
reversal operator, as in Figure 2(a). But the final state has non-zero
momentum, which reverses direction under the time reversal operator,
as in Figure 2(b). The result: the laws of motion for the harmonic os-
cillator are time reversal invariant, and the initial state is preserved by
the time reversal operator, but the final state is not. Curie’s principle
fails when the symmetry is time reversal, in systems as simple as the
harmonic oscillator.

2.2. Mathematical verification. Let’s do the exercise of checking
this result in Hamiltonian mechanics. The possible states of the har-
monic oscillator are the possible values for the position and momentum
(q,p) of the bob in phase space. The laws of motion for the system are
Hamilton’s equations,

d

dt
q(t) =

∂

∂p
h(q,p),

d

dt
p(t) = −

∂

∂q
h(q,p).



4 Bryan W. Roberts

The Hamiltonian h(q,p) for the harmonic oscillator is h(q,p) = (1/2m)q2+
(k/2)p2, where m and k are constants. For simplicity, consider an os-
cillator for which m = 1/2 and k = 2, so that h(q,p) = q2 + p2. The
laws of motion for this system are manifestly time reversal invariant,
in that if (q(t),p(t)) is a possible trajectory, then (q(−t),−p(−t)) is a
possible trajectory as well2.

We now need to check that there is a trajectory with an initial state
that is preserved by time reversal, and a final state that is not. One
such trajectory is the following, which one can check3 is a solution to
the laws of motion above:

q(t) = cos(2t), p(t) = −sin(2t).

At time t = 0, this system has zero momentum, since p(0) = sin(0) = 0.
But it has non-zero momentum for the subsequent times 0 < t < 2π.
The time reversal operator T : (q,p) 7→ (q,−p) therefore preserves the
initial state, but not all later states.

2.3. Summary. Here is what we have observed in the example above:

(1) The harmonic oscillator is time reversal invariant. This is a
simple mathematical fact about the law of motion for the har-
monic oscillator.

(2) We chose a trajectory with an initial state that is preserved by
time reversal. In particular, we chose a trajectory for which
the harmonic oscillator is not always in equilibrium, and then
picked an initial state with zero momentum.

(3) Not all later states of that trajectory are so preserved. The later
states of the harmonic oscillator have non-zero momentum, and
so are not preserved by the time reversal operator.

Curie’s principle thus fails when the symmetry transformation is time
reversal. Let me now say briefly what the origin of this failure is in
mathematical terms.

2To verify: Let (q(t),p(t)) be a solution to Hamilton’s equations. The Hamilton-
ian h(q,p) = p2 + q2 has the property that h(q,p) = h(q,−p). So, Hamilton’s equa-
tions also hold for h(q,−p). But Hamilton’s equations hold for all values of t, and
therefore under the substitution t 7→−t. Making this substitution, we thus find that
−(d/dt)q(−t) = ∂h(q,p)/∂p and hence that (d/dt)q(−t) = ∂h(q,−p)/∂(−p); sim-
ilarly, −(d/dt)p(−t) = −∂h(q,p)/∂q, and hence (d/dt)(−p(−t)) = −∂h(q,−p)/∂q.
That is, (q(−t),−p(−t)) is also a solution to Hamilton’s equations.

3Namely, dq/dt = (d/dt)(cos(2t)) = −2 sin(2t) = 2p(t) = ∂h/∂p, and dp/dt =
(d/dt)(−sin(2t)) = −2 cos(2t) = −2q(t) = −∂h/∂q.



The simple failure of Curie’s principle 5

Hamiltonian mechanics comes equipped with an object called a sym-
plectic form; it can be written dq∧dp in Darboux coordinates. Symme-
try transformations in Hamiltonian mechanics include not only sym-
plectic transformations, which leave the symplectic form invariant, but
also antisymplectic transformations, which reverse its sign. It is easy
to see that the time reversal transformation T(q,p) = (q,−p) is anti-
symplectic, because it reverses the sign of the symplectic form.

Mathematically, the problem for Curie’s principle comes down to
this. A system with a Hamiltonian h might be invariant under a trans-
formation S(q,p) 7→ (q′,p′) that is symplectic or antisymplectic. It
would be standard parlance to call the transformation a “symmetry”
either way. If S is symplectic, then it follows that S-invariant states
evolve to S-invariant states. But this conclusion does not follow when
S is antisymplectic. Time reversing the harmonic oscillator provides
just one simple example of this.

3. Failure in quantum theory

Curie’s principle fails just as badly in quantum theory. Before illus-
trating this, I’ll describe the standard definition of time reversal and
time reversal invariance in the quantum context, since they may be
unfamiliar. Then, rather than redo the harmonic oscillator example
above in the context of quantum theory, I’ll illustrate a general class of
systems (of which the harmonic oscillator is just one example) for which
Curie’s principle fails, and summarize the character of this failure just
as I did for the classical case.

3.1. Time reversal in quantum theory. Curie’s principle fails quite
generally in both non-relativistic quantum mechanics and in relativis-
tic quantum field theory. To keep the discussion general enough to
apply to both, I will characterize the spacetime on which quantum
theory takes place as an affine space M, which admits a foliation into
spacelike hypersurfaces. This will allow us to think of M as either a
non-relativistic spacetime (such as Newtonian or Galilei spacetime), or
a relativistic spacetime (such as Minkowski spacetime).

The vector states of a quantum system will be described by vectors
in a Hilbert space H. For any foliation Σt of the spacetime M into
spacelike hypersurfaces, we take there to be a continuous one-parameter
group of unitary operators Ut = e−itH, which describes how a state
ψ ∈H changes by the rule,

ψ(t) = e−itHψ.



6 Bryan W. Roberts

In differential form, this law becomes the familiar Schrödinger equation
i(d/dt)ψ(t) = Hψ(t), which holds for all ψ(t) in the domain of H.

What do time reversal and time reversal invariance mean in this con-
text? Time reversal in quantum mechanics takes a trajectory ψ(t) to a
new trajectory Tψ(−t), where T : H→H is a bijection called the time
reversal operator. This operator T has the special property of being
antiunitary. Like a unitary operator, an antiunitary operator satisfies
T∗T = TT∗ = I. But unlike a unitary operator it is antilinear, mean-
ing that for any two vectors ψ and φ and for any complex constants a
and b,

(1) T(aψ + bφ) = a∗Tψ + b∗Tφ.

For completeness, let me briefly rehearse one simple argument for
why time reversal has this unusual property. We will make use of a
representation of the canonical commutation relations [Q,P]ψ = iψ,
although such a representation is not necessary, and considerably more
general arguments can be given4. As before, assume that by reversing
time “at an instant,” we preserve positions while reversing momenta:
TQT−1 = Q and TPT−1 = −P . Applying T to both sides of the
commutation relations then immediately implies that,

TiT−1ψ = T[Q,P]T−1ψ = [TQT−1,TPT−1]ψ = [Q,−P ]ψ = −iψ.

No linear operator can reverse the sign of the complex constant i. More-
over, Wigner’s theorem requires that all candidates for symmetry trans-
formations be linear or antilinear. So, since T cannot be linear, it must
be antilinear.

What now does it mean to say that a quantum system is time reversal
invariant? As in classical mechanics, we say that a quantum system
(H,e−itH) is time reversal invariant if and only if, whenever ψ(t) is a
solution to the law of motion, then so is Tψ(−t). This is equivalent5
to the statement,

(2) THT−1 = H,

where H is the generator (the “Hamiltonian”) appearing in the unitary
dynamics Ut = e−itH.

4The original textbook treatment of Wigner (1931, §20) remains one of the best.
Excellent modern treatments have also been given by Sachs (1987, §3.2) and Wein-
berg (1995, §2.6).

5This was pointed out, for example, in (Earman 2002, p.248).



The simple failure of Curie’s principle 7

3.2. Curie’s principle in quantum theory. The quantum harmonic
oscillator provides a counterexample to Curie’s principle just like the
classical harmonic oscillator does. Carrying the example over into the
quantum context is a simple exercise. So, instead of doing this, let me
illustrate a more general class of quantum system (H,e−itH) for which
Curie’s principle fails. This class will include the harmonic oscillator
as a special case. Namely, consider any quantum system that satisfies:

(i) time reversal invariance, THT−1 = H, and
(ii) non-degeneracy, that Hψ1 = hψ1 and Hψ2 = hψ2 only if ψ1 =

eiθψ2 for some θ ∈ [0, 2π].
Non-degeneracy says that if two states have the same value of energy,
then they must be related by a phase-factor eiθ, and thus actually rep-
resent the same probabilistic state6. A Hamiltonian is non-degenerate
if two distinct energy states never occupy the same energy level.

We will show that for the time reversal symmetry transformation,
any system satisfying (i) and (ii) provides a counterexample to Curie’s
principle. We first construct an initial state φ that is preserved by T,
and then exhibit a final state φ(t) that is not preserved by T . The
first step will make use of the following fact about such systems: these
systems generally admit at least two distinct states φ1,φ2 ∈ H such
that Tφ1 = φ1 and Tφ2 = φ2. This fact can be derived by noting that
if ψ1 is an energy eigenvector (Hψ1 = h1ψ1), then by time reversal
invariance,

H(Tψ1) = THψ1 = Th1ψ1 = h1(Tψ1).

That is, Tψ1 is also an eigenstate of H with eigenvalue h1. But since H
is non-degenerate, this implies that Tψ1 = e

iθψ1 for some phase factor
eiθ. Let φ1 := e

iθ/2ψ1. Since T is antilinear, Te
iθ/2 = e−iθ/2T. The

energy eigenstate φ1 thus has the property that,

Tφ1 = Te
iθ/2ψ1 = e

−iθ/2Tψ1 = e
−iθ/2eiθψ1 = e

iθ/2ψ1 = φ1.

By repeating this procedure with an energy eigenvector ψ2 that is or-
thogonal to ψ1, we get a second state φ2 such that Tφ2 = φ2 as well.

The violation of Curie’s principle can now be seen by setting our
initial state φ to be the superposition of the two orthogonal energy

6Vectors related by a phase factor occupy the same ray, defined by a set Ψ :=
{eiθψ : θ ∈ [0, 2π]}. Vectors on the same ray have the same probabilistic content,
in that the transition probability |〈φ,ψ〉|2 remains the same when ψ is replaced
with another vector eiθψ on the same ray (and similarly for φ). It is for this reason
we often say a pure state in ordinary quantum mechanics is best represented by a
ray, rather than a vector.



8 Bryan W. Roberts

eigenvectors φ1 and φ2:

(3) φ = 1√
2

(φ1 + φ2) .

Since Tφ1 = φ1 and Tφ2 = φ2, it follows that Tφ = φ as well. So, our
initial state φ is preserved by the time reversal operator. Indeed, this
initial state is preserved by T in a more general sense as well, in that the
ray Φ = {eiθφ : θ ∈ [0, 2π]} is left unchanged by the transformation
that applies T to each element.

We can now verify that the final state φ(t) is not generally preserved
by T, contrary to the conclusion of Curie’s principle. After a length of
time t, the state φ evolves to

(4) φ(t) = 1√
2

(
e−ith1φ1 + e

−ith2φ2
)
,

where h1 and h2 are the energy eigenvalues of φ1 and φ2, respectively.
Again applying the consequence of antilinearity that Te−ith = eithT ,
we get that,

Tφ(t) = 1√
2

(
Te−ith1φ1 + Te

−ith2φ2
)

= 1√
2

(
eith1φ1 + e

ith2φ2
)
.

This expression is not equal to φ(t) when t ∈ (0, 2π). It does not even
lie on the same ray. So, although T preserves the initial state defined
in Equation (3), it does not generally preserve the final state defined
in Equation (4), in spite of the fact that the system is time reversal
invariant.

Thus, any system that is both time reversal invariant and non-
degenerate provides a counterexample to Curie’s principle. As is well-
known7, the quantum harmonic oscillator satisfies both these condi-
tions, and is therefore just a special case of this general failure.

3.3. Summary. By considering an arbitrary time reversal invariant
system with a non-degenerate Hamiltonian, we have observed:

(1) Such systems are time reversal invariant, by explicit assump-
tion.

(2) We chose a trajectory with an initial state that is preserved by
time reversal. In particular, we constructed a superposition of
energy eigenstates that is preserved by the time reversal oper-
ator T.

(3) Not all later states on that trajectory are so preserved. We
showed that this superposition evolves to states that are not
preserved by T.

7Cf. (Messiah 1999, §XII.4)



The simple failure of Curie’s principle 9

Thus, Curie’s principle fails for quantum mechanics as well. As in
the classical case, the mathematical root of the problem lies in the
special nature of time reversal in quantum theory. This time it is the
antiunitary character of time reversal.

According to standard usage of the term, a transformation is a “sym-
metry” if it is unitary or if it is antiunitary. If a system is invariant
under a symmetry that is unitary, then we can immediately conclude
that S-symmetric states evolve to S-symmetric states. This is because,
for unitary operators, SHS−1 = H implies that Se−itHS−1 = e−itH,
and hence that,

Sψ = ψ ⇒ Sψ(t) = S(e−itHψ) = e−itHSψ = ψ(t).

In other words, Curie’s principle is true for unitary operators! But the
conclusion does not follow for antiunitary operators. In particular, if
T is antiunitary, then THT−1 = H implies that Te−itHT−1 = eitH,
because conjugating the complex number reverses its sign. One can
infer from THT−1 = H that,

Tψ = ψ ⇒ Tψ(t) = T(e−itHψ) = eitHTψ = ψ(−t).

But it is not generally the case that Tψ(t) = ψ(t), as Curie’s principle
would have it. Indeed, this conclusion would only be true in general if
ψ(t) = ψ(−t), which holds only of a trajectory ψ(t) = ψ that remains
unchanged for all time8. So, if a time reversal invariant quantum system
is interesting enough to allow a state that is preserved by time reversal
to change over time, then that system is a counterexample to Curie’s
principle.

4. Three attempts to resist failure

The simple conclusion that I would like to defend is that Curie’s prin-
ciple is not generally true: the principle is formulated for an arbitrary
symmetry, but it is false when that symmetry is time reversal. I would
now like to discuss several true principles that are in the neighborhood
of Curie’s. Given their similarity to the formulation discussed so far,
one might wonder if it is possible to resist the failure of Curie’s prin-
ciple through some clever interpretive maneuvering. I will argue here

8Proof: Let ψ(t) (for t ∈ R) be a trajectory. Suppose that for every choice of ini-
tial state φ on that trajectory, φ(t) = φ(−t) for all t, where φ(t) := e−itHφ. For any
fixed time t, define φ := e−i(t/2)Hψ. Then by assumption φ(t/2) = φ(−t/2). But
φ(t/2) = e−i(t/2)Hφ = e−i(t/2)He−i(t/2)Hψ = ψ(t), while φ(−t/2) = eitHe−itHψ =
ψ. Therefore, ψ(t) = ψ.



10 Bryan W. Roberts

that it is not. There are at least three ways to reinterpret Curie’s prin-
ciple to get a true proposition. None appear to provide a satisfactory
interpretation of the principle.

4.1. Argue time reversal is not a symmetry. One way to rein-
terpret Curie’s principle is to modify what counts as a “symmetry
transformation.” The basic strategy is to say that, in the context of
quantum theory, only unitary transformations can be symmetry trans-
formations, and time reversal is not unitary. This allows one to avoid
the counterexamples altogether, by expelling time reversal (and all an-
tiunitary transformations) from the garden of symmetries.

The strategy can be applied in classical mechanics as well. Curie’s
principle fails for antisymplectic transformations in classical mechan-
ics in just the same way that it fails for antiunitary transformations
in quantum mechanics. So, by requiring that all symmetries be sym-
plectic, we can make the world safe for Curie’s principle in classical
mechanics too.

Earman has formulated a statement of Curie’s principle that might
be seen as adopting this view in the algebraic framework for quantum
field theory. He begins with an algebra of observables, together with an
automorphism group α describing the dynamics. His approach is then
to characterize a “symmetry transformation” in quantum field theory
as a (linear) automorphism θ of the C∗ algebra. In this framework,
Earman writes:

Proposition 2 (Curie’s principle). Suppose that the ini-

tial state ωo is θ-symmetric (i.e. θ̂ωo := ωo◦θ = ωo) and
that the dynamics α is also θ-symmetric (i.e. θαθ−1 =
α). Then the evolved state ω1 := α̂ωo is θ-symmetric.
(Earman 2004, p.198)

This certainly resembles Curie’s principle: the dynamics are determin-
istic (CP1), the dynamics are preserved by a symmetry (CP2), the
initial state is preserved by the symmetry (CP3), and we conclude that
the final state is preserved by the same symmetry (CP4). An eas-
ier special case of this can be formulated in non-relativistic quantum
mechanics. There, the approach would be to characterize a symme-
try transformation θ as a (linear) unitary transformation on a Hilbert
space H. Then we have:

Non-Relativistic Proposition 2. Suppose that the initial
state ψ0 ∈ H is θ-symmetric (i.e. θψ0 = ψ0) and that
the unitary group e−itH generating the dynamics is also



The simple failure of Curie’s principle 11

θ-symmetric (i.e. θe−itHθ−1 = e−itH). Then the evolved
state ψ1 := e

−itHψ0 is θ-symmetric.

Both of these propositions are mathematically correct, and their proofs
are trivial9. Time reversal is excluded from the content of both propo-
sitions because it is antilinear.

Although Earman’s approach results in a Curie-like principle, it is
at the expense of the orthodox definition of symmetry transformations.
In quantum theory, symmetry transformations include not only the
unitary transformations (which are linear), but also anti unitary trans-
formations (which are antilinear). In the algebraic framework in which
Earman works, symmetry transformations include both automorphisms
and anti-automorphisms. This is the orthodox view of symmetries,
arising out of Wigner’s theorem and its generalizations10. In classical
mechanics, the situation is similar: both symplectic and antisymplec-
tic transformations are considered symmetries. So, this approach saves
Curie’s principle only by using standard language in a non-standard
way.

4.2. Argue for a non-standard notion of invariance. A clever
response is to notice that, although Earman’s discussion does not men-
tion antilinear operators, the above two propositions actually do hold
when θ is antilinear. (Their proofs go through in the very same way.)
However, when θ = T is the time reversal operator, the premise corre-
sponding to (CP2) that Te−itHT−1 = e−itH (or θαθ−1 = α in Earman’s
language) does not capture the usual notion of “invariance under time
reversal.” As we saw in Section 3.1, time reversal invariance is equiv-
alent to the statement that THT−1 = H, and since T is antiunitary,
this is equivalent to Te−itHT−1 = eitH. So, time reversal invariance
does not mean that the dynamics is unchanged, but that it is reversed
in time.

But suppose we modify our notion of invariance. One might interpret
“invariance” to mean that e−itH is preserved, and adopt this principle
instead of Earman’s premise (CP2). Then we would have,

(CP2′) The laws of motion/field equations governing the system are
such that if an initial state is invariant under a symmetry,

9Earman states the former; the latter is similar: θψ1 = θe
−itHψ0 = e

−itHθψ0 =
e−itHψ0 = ψ1.

10Cf. (Wigner 1931, §20), (Uhlhorn 1963), (Varadarajan 2007, Theorem 4.29);
the latter two take a symmetry to be an automorphism of the lattice of projec-
tions, which extend to both automorphisms and the anti-automorphisms of the C∗

algebra.



12 Bryan W. Roberts

then so is every final state; in particular, in quantum theory,
Se−itHS−1 = e−itH.

This statement, together with Earmans (CP3) “the initial state of the
system is invariant under said symmetry transformation,” obviously
implies (CP4): “the final state of the sys- tem is also invariant under
said symmetry transformation” by simple modus ponens. In particular
we can verify in quantum theory that if Sψ = ψ and Se−itHS−1 =
e−itH, then Sψ(t) = ψ(t). So we have another true statement that
resembles Curie’s principle.

Earman may have had this condition in mind, in suggesting the
equivalent statement that we “understand the invariance of laws of
motion/field equations to mean that if an initial state is evolved for any
chosen ∆t to produce a final state and then the symmetry operation
is applied to the final state, the resulting state is the same as obtained
by first applying the symmetry operation to the initial state and then
evolving the resulting state for the same ∆t” (Earman 2004, p.176). On
this reading of “invariance,” Earman’s Proposition 2 provides a correct
statement, which resembles Curie’s principle, and which applies to both
linear and antilinear operators.

The price of this response is that one must give up the standard
meaning of “time reversal invariance,” in favor of a property that is
almost never satisfied. The non-standard interpretation of time rever-
sal invariance holds whenever Te−itHT−1 = e−itH. But the standard
interpretation of time reversal invariance is that Te−itHT−1 = eitH. It
is easy to verify11 that a system can only be simultaneously time re-
versal invariant in both the standard and the non-standard senses if
the dynamics is trivial, e−itH = I. So, since almost all known quantum
systems satisfy the standard definition of time reversal invariance, it
follows that almost none of them satisfy the non-standard definition.
In other words, the price of this approach is really to render Curie’s
principle inapplicable to almost every quantum system.

4.3. Argue that Curie’s principle is about trajectories. A third
response is to retain the orthodox definitions of symmetry and invari-
ance but to modify the kind of object that Curie’s principle is about.
The last premise and the conclusion of Curie’s principle (Earman’s CP3
and CP4) are about states. They read:

(CP3) the initial state of the system is invariant under the symmetry;
(CP4) the final state of the system is invariant under said symmetry.

11If Te−itHT−1 = e−itH and Te−itHT−1 = eitH, then e−itH = eitH. But Stone’s
theorem implies that this group has a unique generator, so H = −H. This is only
possible if H = 0, and hence e−itH = I.



The simple failure of Curie’s principle 13

But premise (CP2) is about invariance of the laws, which on the stan-
dard interpretation refers to an entire trajectory. In particular (as
discussed in Section 3.1), the laws are invariant under a transforma-
tion if whenever ψ(t) is a possible trajectory, then so is the transformed
trajectory ψ′(t′). So, we can view the trouble with Curie’s principle as
one of discord between two objects interest: states in one premise, and
trajectories in another.

One can bring these objects of interest into closer agreement by mak-
ing all the premises of Curie’s principle about trajectories. To do this,
let us write {ψ(t) = e−itHψ : t ∈ R} to denote the trajectory with
initial state ψ. We begin by distinguishing two senses in which a state
ψ(t) in that trajectory can be “symmetric” with respect to a symmetry
transformation.

(1) A state ψ(t) at a time t is S-symmetric in the original order if
Sψ(t) = ψ(t).

(2) A state ψ(t) at a time t is S-symmetric in the reverse order if
Sψ(t) = ψ(−t).

This is not such an unusual distinction, when one recalls (from the end
of Section 4.1) that the standard definition of time reversal invariance
entails a similar reversal of sign: Te−itHT−1 = eitH.

We can now express a revision of Curie’s principle: If,

(CP1) the laws of motion/field equations governing the system are
deterministic;

(CP2) the laws of motion/field equations governing the system are
invariant under a symmetry transformation; and

(CP3′) the state of the system at some fixed time t0 is symmetric under
said symmetry (in the original or reverse order);
then,

(CP4′) the state of the system at any time t is symmetric under said
symmetry (in the same order).

In the context of ordinary quantum mechanics, this statement corre-
sponds to the following two facts12.

Fact 1. Suppose a state ψ(t0) := e
−it0Hψ at a fixed time t0 is θ-

symmetric in the original order (i.e. θψ(t0) = ψ(t0)), and that the
unitary group e−itH generating the dynamics is invariant under θ in
the original order (i.e. θe−itHθ−1 = e−itH). Then for all times t, the
state ψ(t) = e−itHψ is θ-symmetric in the same order.

12Fact 1 follows from the non-relativistic version of Proposition 2 in the last
subsection. Fact 2 is proved: Tψ(t) = Te−i(t−t0)He−it0Hψ = Te−i(t−t0)Hψ(t0) =
ei(t−t0)HTψ(t0) = e

i(t−t0)ψ(−t0) = ei(t−t0)Heit0Hψ = eitHψ = ψ(−t).



14 Bryan W. Roberts

Fact 2. Suppose a state ψ(t0) := e
−it0Hψ at a fixed time t0 is θ-

symmetric in the reverse order (i.e. θψ(t0) = ψ(−t0)), and that the
unitary group e−itH generating the dynamics is invariant under θ in
the reverse order (i.e. θe−itHθ−1 = eitH). Then for all times t, the state
ψ(t) = e−itHψ is θ-symmetric in the reverse order.

We have again arrived at a correct mathematical statement. Time
reversal is no longer excluded, being captured now by Fact 2. We have
moreover retained the usual definition of a “symmetry/invariance” of
the laws. But is this Curie’s principle? Strictly speaking, Curie’s prin-
ciple says that if the initial state is preserved by a symmetry transfor-
mation, then so is the final state. This is not what is described by Fact
2 above, where the symmetry transformation “flips” each state about
the temporal origin. Facts 1 and 2 perhaps express a more natural
principle, in bringing the premises into closer alignment. But they do
not capture the original expression of Curie’s principle.

5. Conclusion

If one reads “symmetry” in the statement of Curie’s principle to
include time reversal symmetry, then Curie’s principle is false. In par-
ticular, when we try to apply it to time reversal, Curie’s principle fails
for systems as elementary as the harmonic oscillator. It fails in the con-
text of Hamiltonian mechanics, and it fails in the context of quantum
theory.

There remain at least three statements in the neighborhood of Curie’s
principle that are mathematically correct. They can be achieved either
by excluding symmetry transformations like time reversal, or by mod-
ifying the statements (CP2)-(CP4) appearing in the principle. But
although these modifications may be of independent interest, I do not
see that any one provides a plausible way to interpret Curie’s principle.
The correct conclusion, I submit, is that Curie’s principle simply fails
for time reversal symmetry.

References

Brading, K. and Castellani, E. (2003). Symmetries in physics: philo-
sophical reflections, Cambridge: Cambridge University Press.

Curie, P. (1894). Sur la symétrie dans les phénomènes physique,
symétrie d’un champ électrique et d’un champ magnétique, Jour-
nal de Physique Théorique et Appliquée 3: 393–415.

Earman, J. (2002). What time reversal is and why it matters, Interna-
tional Studies in the Philosophy of Science 16(3): 245–264.



The simple failure of Curie’s principle 15

Earman, J. (2004). Curie’s Principle and spontaneous symmetry break-
ing, International Studies in the Philosophy of Science 18(2-3): 173–
198.

Ismael, J. (1997). Curie’s Principle, Synthese 110(2): 167–190.
Messiah, A. (1999). Quantum Mechanics, Two Volumes Bound as One,

New York: Dover.
Mittelstaedt, P. and Weingartner, P. A. (2005). Laws of Nature,

Springer-Verlag Berlin Heidelberg.
Sachs, R. G. (1987). The Physics of Time Reversal, Chicago: Univer-

sity of Chicago Press.
Uhlhorn, U. (1963). Representation of symmetry transformations in

quantum mechanics, Arkiv för Fysik 23: 307–340.
Varadarajan, V. S. (2007). Geometry of Quantum Theory, New York:

Springer Science and Business Media, LLC.
Weinberg, S. (1995). The Quantum Theory of Fields, Vol. 1: Founda-

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(1959).


	1. Introduction
	2. Failure in classical mechanics
	2.1. In pictures
	2.2. Mathematical verification
	2.3. Summary

	3. Failure in quantum theory
	3.1. Time reversal in quantum theory
	3.2. Curie's principle in quantum theory
	3.3. Summary

	4. Three attempts to resist failure
	4.1. Argue time reversal is not a symmetry
	4.2. Argue for a non-standard notion of invariance
	4.3. Argue that Curie's principle is about trajectories

	5. Conclusion
	References