Positive Curvature Can Mimic a Quantum

Abstract

We elaborate on the existing idea that quantum mechanics is an emergent phenomenon, in the form of a coarse-grained description of some underlying deterministic theory. We apply the Ricci flow as a technical tool to implement dissipation, or information loss, in the passage from an underlying deterministic theory to its emergent quantum counterpart. A key ingedient in this construction is the fact that the space of physically inequivalent quantum states (either pure or mixed) has positive Ricci curvature. This leads us to an interesting thermodynamical analogy of emergent quantum mechanics. Comment: 8 pages

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arXiv:0912.1535v1 [hep-th] 8 Dec 2009
POSITIVE CURVATURE CAN MIMIC A QUANTUM
J.M. Isidro1,a,J.L.G. Santander2,b and P. Fern´
andez de C´
ordoba1,c
1Instituto Universitario de Matem´atica Pura y Aplicada,
Universidad Polit´ecnica de Valencia, Valencia 46022, Spain
2Departamento de Ciencias Experimentales y Matem´aticas,
Universidad Cat´olica de Valencia, Valencia 46002, Spain
ajoissan@mat.upv.es,bjlgonzalez@mat.upv.es,
cpfernandez@mat.upv.es
Abstract We elaborate on the existing idea that quantum mechanics is an emergent
phenomenon, in the form of a coarse–grained description of some underlying deter-
ministic theory. We apply the Ricci flow as a technical tool to implement dissipation,
or information loss, in the passage from an underlying deterministic theory to its emer-
gent quantum counterpart. A key ingedient in this construction is the fact that the space
of physically inequivalent quantum states (either pure or mixed) has positive Ricci cur-
vature. This leads us to an interesting thermodynamical analogy of emergent quantum
mechanics.
1 Introduction
Quantum mechanics as a statistical theory has been argued to emerge from an underly-
ing deterministic theory [1]. Specifically, for any quantum system there exists at least
one deterministic model that reproduces all its dynamics after prequantisation. This ex-
istence theorem has been extended to include cases characterised by sets of commuting
beables [2]; it has also been complemented withan explicit dynamical theory [3].
Mechanisms have been presented [1, 2, 3] to explain the passage from a determin-
istic theory to a probabilistic theory. Usually they are based on a dynamical system,
the phase–space trajectories of which possess suitably located attractors (e.g., at the
eigenvalues of the given quantum Hamiltonian, or at certain configurations of the den-
sity matrix). These mechanisms can be thought of as an existence theorem, in that
every quantum system (with a finite–dimensional Hilbert space) possesses at least one
deterministic system underlying it.
On the other hand there are plenty of dissipation equations in physics and mathe-
matics, equations implementing the information loss that is characteristic of the pas-
sage from classical to quantum. The heat equation immediately comes to mind.
In this contribution we develop a deterministic model exhibiting dissipation, from
which quantum mechanics emerges naturally. Given a quantum mechanics with a com-
plex d–dimensional Hilbert space, the Lie group SU (d)represents classical canonical
transformations on the projective space CPd−1of quantum states. Let Rstand for the
Ricci flow [4] of the manifold SU (d−1) down to one point, and let Pdenote the pro-
jection from the Hopf bundle onto its base CPd−1. Then the underlying deterministic
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model we propose here is the Lie group S U (d), acted on by the operation P R.
We would like to mention that additional quantum–mechanical applications of the
Ricci flow havebeen reported in [5, 6, 7, 8]; deterministic models of quantum mechan-
ics and closely related topics are dealt with at length in [9, 10, 11, 12].
2 The Ricci flow as a (nonlinear) heat flow
Given an n–dimensional manifold Mendowed with the Riemannian metric gij , the
equation governing the (unnormalised) Ricci flow reads
∂gij
∂t =−2Rij , i, j = 1,...,n, t ≥0,(1)
where tis an evolution parameter (not a coordinate on M), and Rij is the Ricci tensor
corresponding to the metric gij . Informally one can say that Ricci–flat spaces remain
unchanged under the flow, while positively curved manifolds contract and negatively
curved manifolds expand under the flow. We will be interested in the particular case of
Einstein manifolds, where the Ricci tensor and the metric are proportional:
Rij =κgij,(2)
with κa constant. Since the metric gij is assumed positive definite, the sign of κ
equals the sign of the Ricci tensor. Relevant examples of positively curved Einstein
manifolds are complex projective space CPNand the special unitary group SU(N),
both of which will play an important role in what follows. Their respective metrics are
the Fubini–Study metric [13] and the Killing–Cartan metric [14].
Under the Ricci flow, the contraction of a whole manifold down to a point can
play the role of a dissipative mechanism. One hint that this intuition is correct comes
from the following example. Consider a 2–dimensional manifold endowed with the
isothermal coordinates xand y. Then the metric reads
ds2= e−f(x,y)dx2+ dy2.(3)
Allowing the metric to depend also on the evolution parameter t, the Ricci flow equa-
tion (1) becomes ∂f
∂t =∇2f. (4)
The above is formally identical to the heat equation, with one important difference,
however: the Laplacian ∇2is computed with respect to the metric (3), in which it
reads
∇2f= ef∂2f
∂x2+∂2f
∂y2.(5)
Regardless of the nonlinearity of (5), the fact that the Ricci–flow equation can be recast
as a generalisation of the heat equation is a clear hint that a dissipative mechanism is at
work.
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3 The deterministic model for pure states
In this section we will consider a quantum system with a finite, complex d–dimensional
Hilbert space of quantum states, that we can identify with Cd. Let Cdenote the phase
space of the classical model, the quantisation of which gives the quantum system un-
der consideration. For our purposes the precise nature of this classical model on C
is immaterial. Now unitary transformations on Hilbert space are the quantum coun-
terpart of canonical transformations on classical phase space C. We may thus regard
SU (d)as representing classical canonical transformations, Cdbeing the carrier space
of this representation. We are considering, as in ref. [1], the simplified case of a finite–
dimensional Hilbert space. Without loss of generality we will restrict to those canonical
transformations that are represented by unitary matrices with determinant equal to 1.
Now quantum states are unit rays rather than vectors, so in fact the true space of
inequivalent quantum states is the complex projective space CPd−1. The latter can be
regarded as a homogeneous manifold:
CPd−1=SU (d)
SU (d−1) ×U(1).(6)
In this picture we have SU (d)as the total space of a fibre bundle with typical fibre
SU (d−1) ×U(1) over the base manifold CPd−1. The projection map
π:SU (d)−→ CPd−1, π(w) := [w](7)
arranges points w∈SU (d)into SU (d−1) ×U(1) equivalence classes [w].
Classical canonical transformations as represented by SU (d)act on the Hilbert
space Cd. This descends to an action αof SU (d)on CPd−1as follows:
α:SU (d)×CPd−1−→ CPd−1, α (u, [v]) := [uv].(8)
Here we have u∈SU (d),[v]∈CPd−1, and uv denotes d×dmatrix multiplication.
One readily checks that this action is well defined on the equivalence classes under
right multiplication by elements of the stabiliser subgroup SU(d−1) ×U(1). This
allows one to regard quantum states as equivalence classes of classical canonical trans-
formations on C. Physically, uin (8) denotes (the representative matrix of) a canonical
transformation on C, and [v]denotes the equivalence class of (representative matrices
of) the canonical transformation vor, equivalently, the quantum state |vi.
In the picture just sketched, two canonical transformations are equivalent whenever
they differ by a canonical transformation belonging to SU (d−1), and/or whenever they
differ by a U(1)–transformation. Modding out by U(1) has a clear physical meaning:
it is the standard freedom in the choice of the phase of the wavefunction corresponding
to the matrix v∈SU(d). Modding out by SU (d−1) also has a physical meaning:
canonical transformations on the (d−1)–dimensional subspace Cd−1⊂Cdare a
symmetry of v. Therefore the true quantum state |viis obtained from v∈SU(d)after
modding out by the stabiliser subgroup SU (d−1) ×U(1).
We conclude that this picture contains some of the elements identified as respon-
sible for the passage from a classical world (canonical transformations) to a quantum
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world (equivalence classes of canonical transformations, or unit rays within Hilbert
space). This is so because some kind of dissipative mechanism is at work, through the
emergence of orbits, or equivalence classes. However the projection (7) is an on/off
mechanism. Instead, one would like to see dissipation occurring as a flow along some
continuous parameter. To this end we need some deterministic flow governed by some
differential equation.
We claim that we can render the projection (7) a dissipative mechanism governed
by some differential equation. This equation will turn out to be the Ricci flow (1).
Proof of this statement follows.
The Lie group S U (d−1) ×U(1) is compact, but it is not semisimple due to
the Abelian factor U(1). Leaving the U(1) factor momentarily aside, SU (d−1) is
semisimple and compact. As such it qualifies as an Einstein space with positive scalar
curvature with respect to the Killing–Cartan metric [14]. Now eqn. (1) ensures that
SU (d−1) contracts to a point under the Ricci flow.
However the U(1) factor renders SU (d−1) ×U(1) nonsemisimple. As a conse-
quence, the Killing–Cartan metric of S U (d−1) ×U(1) has a vanishing determinant
[14]. The Ricci flow can still cancel the SU(d−1)–factor within S U (d), but not
the U(1) factor. After contracting SU(d−1) to a point we are left with the space
U(1) ×CPd−1or, more generally, with a U(1)–bundle over the base manifold CPd−1.
This U(1)–bundle over CPd−1is the Hopf bundle, where the total space is the sphere
S2d−1in 2d−1real dimensions [13]. This sphere falls short of being the true space of
quantum states by the unwanted U(1)–fibre, that cannot be removed by the Ricci flow.
It can, however, be done away with by projection Pfrom the total space of the bundle
down to its base. The combined operation “Ricci flow R, followed by projection P”
acts on the stabiliser subgroup SU (d−1) ×U(1) of the initial SU(d)and leaves us
with CPd−1as desired. Therefore this combined operation P R acts in the same way as
the projection πin (7). As opposed to the latter, however, this combined operation P R
provides us with a differential equation that implements dissipation along a continuous
parameter, at least along most of the way.
4 The deterministic model for mixed states
We have so far dealt only with pure quantum states. In trying to extend our previous
analysis to mixed quantum states, we must first answer the following two questions:
What is the manifold of mixed quantum states, and what sign does its Ricci scalar
have? Mixed states can be represented by density matrices D, expressible as
D=
n
X
j=1
|vjipjhvj|, n > 1,(9)
the pj>0being the probability of finding the system in the pure state |vji. Above we
assume that n > 1,i.e., that the state considered is not pure but truly mixed. The pj
must add up to unity, n
X
j=1
pj= 1.(10)
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It turns out that the manifold of density matrices is a norm–closed (with respect to the
trace norm), convex subset of the unit sphere of the space of trace–class operators (see,
e.g., [15]). In the finite–dimensional setup considered here, all operators are trace class,
and we are left with a convex subset of the unit sphere of the space of d×dHermitian
matrices. Now the space of d×dHermitian matrices has (real) dimension d2−1, so
its unit sphere is d2−2(real) dimensional. The manifold of mixed states is a convex
subset of the real sphere Sd2
−2. In particular, the latter has positive Ricci curvature.
On the other hand, the N–dimensional sphere SNequals the homogeneous manifold
SO(N+ 1)/SO(N), so our manifold of mixed quantum states is a convex subset of
Sd2
−2=SO(d2−1)
SO(d2−2).(11)
As in the case of pure states, the special orthogonal groups in the numerator and in
the denominator above carry positive Ricci curvature with respect to the corresponding
Killing–Cartan metric [14].
Now eqn. (11) differs very little from (6), that we discussed at length in the case of
pure quantum states. One difference between these two equations is that (6) contains
the unitary groups, while (11) contains the special orthogonal groups. Another differ-
ence is that the Abelian factor U(1) in the denominator of (6) has disappeared from
(11) (one could still mod out the right–hand side of (11) by the discrete group Z2in
order to obtain real projective space, but the latter is not related to the space of density
matrices). One final difference between the pure and the mixed case is that the latter
does not have the full left–hand side of (11) as the space of quantum states, but only a
convex subset thereof.
All this notwithstanding, these three differences do not suffice to prevent the anal-
ysis (and the ensuing conclusions) of the case of pure quantum states from applying to
the case of mixed quantum states as well. It is interesting to observe that mixed quan-
tum states are actually simpler to deal with than pure states, because the absence of the
Abelian factor in the denominator of (11) allows one to dispense with the projection P
from the Hopf bundle—in fact, in the mixed case there is no Hopf bundle at all.
Having seen that the case of mixed quantum states does not differ substantially
from that of pure states, for the rest of this contribution we will concentrate on pure
quantum states.
5 Positive curvature mimics a quantum
Our starting point was the observation that canonical transformations on classical phase
space are implemented quantum–mechanically as unitary transformations on the Hilbert
space of quantum states. In our finite–dimensional setup, this gave rise to a natural ac-
tion of SU(d)on Cd. This action provided us with the building blocks to construct
the deterministic system that we take to underlie the given quantum mechanics. Next,
different pieces of classical information (elements of SU(d), or classical canonical
transformations) were arranged into quantum equivalence classes (points on CPd, or
quantum states): this procedure implements information loss, or dissipation. Quantum
states thus arose as equivalence classes of canonical transformations on classical phase
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space. However, dissipation was not implemented by means of the usual projection
(7) (an on/off mechanism), but rather by means of the Ricci flow (followed by the pro-
jection P). The rationale was that the Ricci flow provided us with a a deterministic
mechanism governed by a dissipative differential equation, that can be understood as a
flow along a continuous parameter.
In a nutshell, our deterministic model is the group manifold SU(d), acted on by
the combined operation P R described above. Here Rstands for the Ricci flow of
SU (d−1) down to one point, and Pstands for the projection from the Hopf bundle
with total space S2d−1onto its base CPd−1.
The previous conclusions can be compactly recast, somewhat in the style of news-
paper headlines, as positive curvature mimics a quantum [16, 17]. This raises the
question, how about negative curvature? From what was said above it should be clear
that the answer is negative: negative curvature cannot mimic a quantum, since negative
curvature causes expansion, rather than contraction to a point. Temporarily abandoning
our geometrical stance, let us take a brief thermodynamical detour that will lend further
support to our statement concerning negative curvature. Thermodynamics is a coarse–
grained version of an underlying microscopic theory, namely, statistical mechanics.
Coarse–graining the notion of mechanical energy in statistical mechanics gives rise to
the thermodynamical notion of heat. In our setup, the analogue of the heat equation
is the Ricci flow equation. Admittedly, the classical heat equation is linear while the
Ricci flow equation is not, but this difference will play no role here. In fact, the analogy
between these two equations goes so far, that the Ricci flow equation (1) can actually
be derived from a functional F, called Perelman’s functional [4], which happens to be
a monotonically increasing function of the time t. Therefore Fqualifies as an entropy.
Now heat flow occurs from higher temperature to lower temperature; such a heat flow
is accompanied by an increase in entropy. This reveals what the thermodynamical ana-
logue of negative curvature must be: heat flowing from lower temperature to higher
temperature. This is clearly unphysical. Thus thermodynamics meets geometry in the
statement that negative curvature cannot mimic a quantum.
We would also like to point out that a related form of coarse–graining has been
put forward [18] in order to explain the emergence of quantum mechanics from an
underlying deterministic theory. In fact we can provide a dictionary between our ther-
modynamical analogy, on the one hand, and the requirements imposed on the operation
of coarse–graining in ref. [18], on the other. Namely: probability conservation in [18]
corresponds to energy conservation in our thermodynamical analogy, while dissipation
in [18] is matched, in our picture, by increase in entropy.
Looking beyond, one could even pose the question: regardless of the sign, why
curvature at all? We have seen that Ricci–flat spaces remain unchanged under the Ricci
flow. In our thermodynamical analogy, this would correspond to no heat flow at all,
that is, to a temperature distribution satisfying the static Laplace equation ∇2T= 0. A
nonvanishing (and, as we have argued, positive) value of the scalar curvature provides
us with a natural length scale: the Ricci scalar. With it, a natural notion of a quantum
comes along.
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Acknowledgements J.M.I. is pleased to thank the organizers of the meeting QTRF5
(Quantum Theory: Reconsideration of Foundations 5, V¨axj¨o, Sweden) for the invita-
tion to participate, and for providing a congenial environment for scientific exchange.
It is also a great pleasure to thank H.-T. Elze and B. Koch for discussions. This work
has been supported by Universidad Polit´ecnica de Valencia under grant PAID-06-09.—
Once you’ve reached the top, don’t forget to send the elevator down, for the next guy.
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