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Entropy 2015, 17(8), 5888-5902; doi:10.3390/e17085888
Abstract: We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalized complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields on the Schroedinger wavefunction.
The theory of thermodynamical fluctuations provides a solid link between macroscopic and microscopic physics. Classical fluctuation theory  often sheds light on counterintuitive quantum-mechanical phenomena, thus helping to bridge the gap between the classical world and the quantum world. For example, Heisenberg’s uncertainty principle can be nicely illustrated resorting to the theory of Gaussian fluctuations around thermal equilibrium .
On the other hand, the theory of thermodynamical fluctuations can be recast using the geometric language of differential manifolds [3,4,5,6,7,8,9]. This reexpression of a physical discipline in more abstract mathematical language goes a long way beyond a mere rewriting of the concepts involved. It renders the theory more versatile, enlarging its scope. Moreover, since the advent of Einstein’s general relativity a century ago, (pseudo) Riemannian geometry belongs to the technical skills that any physicist has to master (at least at a working level). This places (pseudo) Riemannian geometry at a vantage point. In the opposite direction (i.e., thermodynamics as applied to geometry) one should mention at least two developments. The first one is a whole body of knowledge on the thermodynamics of black holes [10,11]. More recently, the reexpression of Einstein’s relativity as a thermodynamics [12,13] has had far-reaching consequences for our understanding of spacetime.
In trying to understand the thorny relationship between gravity and the quantum [16,17,18,19] it has been argued that gravity acts dissipatively on quantum systems . Specifically, in the presence of a gravitational field, thermal fluctuations become indistinguishable from quantum fluctuations [21,22,23]. This raises the fundamental question: How is one to treat thermal and quantum fluctuations on the same footing? Is it altogether possible? We will see here that generalized complex manifolds provide one viable answer to this question, one that appears not to have been explored yet in the geometrical approach to thermodynamics.
2. Geometry and Fluctuations
2.1. Riemannian Geometry
As a very elementary example, consider a thermodynamical system in an equilibrium state described by the following variables: temperature T, pressure P and volume V. In the Gaussian approximation, choosing T and V as independent variables, the probability W of a fluctuation , around equilibrium is given by 
2.2. Symplectic Geometry
As our starting point here we will consider a certain thermodynamical system in equilibrium, in order to arrive at a corresponding symplectic structure.
Again in the Gaussian approximation, the probability W of a fluctuation , , , is given by 
The importance of symplectic structures in classical mechanics is widely recognized and need hardly be recalled . In fact not just Riemannian geometry, but also symplectic geometry, pertains to the realm of thermal fluctuations: the first law of thermodynamics endows the thermodynamic phase space with a contact structure, which includes symplectic geometry as a sub-case [3,4,6,7].
A real -dimensional manifold M is symplectic if there exists a closed, non-degenerate, rank 2 antisymmetric tensor field defined everywhere on M. Let be local coordinates around , so with . Since the matrix is nonsingular, an inverse exists such that . The Poisson brackets of two functions are defined as , and the integrability condition turns out to be equivalent to the Jacobi identity for these Poisson brackets.
In this way the following symplectic analogue of Equation (3) allows one to compute the average value of the function f on M:
One can also regard a symplectic structure as providing an isomorphism from the tangent space into the cotangent space at each . Specifically, the tangent vector is mapped into the 1-form , with . This viewpoint motivates the following definition (equivalent to the above, but more useful for later applications): a symplectic structure over a -dimensional manifold M is an isomorphism between the tangent and the cotangent fibers over each point ,
2.3. Complex Geometry and Kähler Geometry
Informally one could say that the imaginary unit is the hallmark of quantum mechanics. That pertains to the quantum world has been very interestingly argued recently in [26,27]. More standard arguments have been known for long; such are the heat equation in imaginary time , or the fact that quantum commutators formally equal times classical Poisson brackets . Here we will briefly recall the role played by complex structures in the theory of coherent states [28,29].
Let M be a real -dimensional phase space endowed with the symplectic form ω. For simplicity let us also assume that M admits a holomorphic atlas compatible with the symplectic structure (this compatibility condition is called the Kähler property). In plain words, the real and imaginary parts of the holomorphic coordinates are Darboux coordinates for ω (here assumed dimensionless for simplicity):
Every complex manifold M admits a (positive definite) Hermitian metric that is compatible with the complex structure . Then an analogue of Equations (3) and (11) gives us the average value of a function f on M:
Formally, a complex structure J over a real -dimensional manifold M is an endomorphism of the tangent fibre over each point
2.4. Generalized Complex Geometry
Our original motivation was the statement [21,22,23] that, in the presence of a gravitational field, quantum fluctuations become indistinguishable from thermal fluctuations. We have argued that thermal fluctuations are associated with symplectic structures, while quantum fluctuations come along with complex structures. How, then, is one to treat thermal and quantum fluctuations on the same footing? This is trivially achieved by those phase spaces M that qualify as Kähler manifolds. However, the Kähler condition is very restrictive: not only does M have to be simultaneously complex and symplectic; these two independent structures also have to be compatible.
In references. [3,4] the geometry of the thermodynamic phase space (including fluctuations) results in a para-Sasakian manifold, which is the contact-geometry equivalent of a Kähler manifold in symplectic geometry. This means that if one restricts to a proper even-dimensional subspace, the geometry is indeed that of a Kähler manifold. This geometry achieves the goal of treating thermal and quantum fluctuations on the same footing.
Generalized complex structures (GCS) also achieve the goal of providing a unified framework for thermal and quantum fluctuations. The following is a brief summary of GCS extracted from , duly tailored to meet our needs. For simplicity we prefer to work locally around a point . Global issues can be taken care of by the corresponding integrability conditions, to be mentioned along the way whenever necessary. For our purposes the -dimensional manifold M is assumed to be a phase space, that is, , for a certain n-dimensional configuration space S.
Rather than considering the fibres or separately, in generalized complex geometry one considers their direct sum: over each point one erects the fibre . The total space of the bundle so constructed is -dimensional: dimensions for the base M, dimensions for the fibre.
An inner product is defined on the fibre :
The particular isometries of the fibre obtained by setting in Equation (22) and exponentiating,
A generalized complex structure over M, denoted J, is an endomorphism of the fibre over each ,
Furthermore, GCS succeed at interpolating between the above opposite types, the symplectic type and the complex type; let us explain this more carefully. A point is said to be regular if it possesses a neighborhood on which there exists a Poisson structure with constant rank. In a neighborhood of any regular point one can define a diffeomorphism and a B-transformation, the combined action of which maps into the product . Here is an open set within the standard complex manifold , and is an open set within the standard symplectic manifold . The nonnegative integer k is called the type of the GCS J, the limiting cases of Equations (30) and (31) respectively corresponding to and . As described in [14,15], the type k need not be constant across M: it may vary from one point to another in M.
In plain words, any generalized complex manifold factorizes locally as the product of a complex manifold times a symplectic manifold.
Finally assume that M is a linear space. Then any generalized complex structure of type is the B-transform of a symplectic structure. This means that any generalized complex structure of type can be written as
The consequences of the above become immediately apparent. Let us for simplicity assume that the type k is constant across M. Then any GCS with an extremal value of k, i.e., either or , can always be reduced to the corresponding canonical form (30) or (31) by means of a B-transformation. Thus corresponds to a thermal description of phenomena, while corresponds to a quantum description of phenomena, no interpolation existing between the two descriptions. Nonextremal values of the type, i.e., such that , contain both thermal and quantum descriptions simultaneously.
3. When “Quantum” Becomes “Thermal”
Any gravitational field is locally equivalent to an accelerated frame. In an accelerated frame, quantum becomes thermal; this is basically the content of the Unruh effect  (in an admittedly lax formulation that is however precise enough for our purposes). Without using the full apparatus of relativistic quantum field theory, let us see how quantum can become thermal in the simplified setup of the quantum mechanics of a nonrelativistic particle. This understood, we will analyse the role played by the GCS on phase space under the passage from an inertial frame to an accelerated frame. We will conclude that the transformation law for the Schroedinger wavefunction under the passage to a noninertial frame (as in the Unruh effect) is governed by a B-transformation of the GCS on phase space.
A remark is in order. The gravitational field considered here must be weak in order to rule out effects such as, e.g., relativistic speeds, or the likely breakdown of standard quantum mechanics in the presence of very strong gravitational fields . Such phenomena lie beyond our scope.
3.1. Inclusion of a Gravitational Field
In flat Euclidean space , let K denote an inertial frame with origin O and axes , and . Let denote a uniformly accelerated frame, with origin and axes , and respectively parallel to , and . For simplicity we will assume that, at , the two origins O and coincide, their relative velocity also vanishing at . Let the acceleration of with respect to K be , with α a constant. Coordinates with respect to K are related to coordinates with respect to as per
3.2. The Unruh Effect
The next step is to invoke de Broglie  in order to write an inverse proportionality between time t and temperature T:
The Boltzmann-like factor present in Equation (43) bears out the fact that the effect of the gravitational field on the Schroedinger wavefunction is of thermal nature. Due to the assumptions made in our derivation, Equation (43) is valid only for intermediate temperatures. The limit is excluded (because this would require strong gravitational fields); so is the limit (because of the inverse proportionality Equation (40) between time and temperature).
3.3. Transformation to an Accelerated Frame as a B-Transformation
Classical phase space is spanned by the coordinates and their conjugate momenta . For the rest of the discussion, the dimensions y, , z, can be ignored, as they are unaffected by the change of frame Equation (34). Thus, for our purposes, the manifold M of Section 2.4 can be taken to be that subspace of classical phase space spanned by x and , i.e., .
Now the manifold can be endowed with a GCS. This can be done in two equivalent ways. One can consider the GCS of complex type defined on by the complex coordinates Equation (14). Alternatively, one can consider the GCS of symplectic type defined on by the standard symplectic form . Since our interest lies in considering the effect of B-transformations, and is a Kähler manifold, the type of the CGS considered is immaterial.
We claim that the transformation law for the Schroedinger wavefunction under the passage to an accelerated frame, Equation (39), follows from a B-transformation of the GCS on phase space , Equation (26). In other words, the Schroedinger wavefunction keeps track of which frame is being used, the bookkeeping device being the GCS on phase space. Verifying that such is indeed the case requires, so to speak, translating the geometer’s language into the physicist’s language. This we do next.
Tangent vectors X at the point are objects
Next we reexpress the B-transformation (26) as the variation
Along the motion of the particle located at we can write, using Equation (34),
Summarising, we may say that the Unruh effect acts on the wavefunction by multiplication with the exponential of ( times) the integral of a B-field along the particle’s trajectory on phase space. The vector field X involved in this B-transformation is just the position vector on phase space, while the B-field considered is a mere scalar multiple of the canonical symplectic form on phase space.
3.4. A Nonuniform Gravitational Field
The relation just derived between the Unruh effect and the B-transformation of the GCS on phase space was based on the assumption that the gravitational field was static and spatially constant. In turn, this assumption made it possible to choose a constant B-field on phase space (actually a scalar multiple of the symplectic form). A nonstatic and/or nonuniform gravitational field can be mimicked by a nonstatic and/or nonuniform acceleration vector . This lends plausibility to the following hypothesis:
Hypothesis 1. Regard classical phase space as a generalized complex manifold. In the presence of a nonstatic and/or nonuniform, but nevertheless weak, gravitational field, the inertial-frame Schroedinger wavefunction ψ remains form-invariant under a transformation to a locally accelerated frame, where its value is , provided that ψ and are related according to the law
We defer analysis of the above hypothesis for further study.
We have presented a brief review of some recent developments in differential geometry with applications to thermodynamical fluctuation theory. Standard wisdom draws a clear frontier between thermal fluctuations and quantum fluctuations. While this separation is perfectly consistent in the absence of gravitational fields, this border becomes fuzzy in the presence of gravity [20,21,22,23]. A well-known example of this mixing is the Unruh effect [31,33,34]. Another instance of a gravitational incursion into the thermal realm is the Ehrenfest-Tolman effect . One can expect an eventual theory of quantum gravity to enhance, rather than diminish, this mixing of thermal and quantum phenomena.
In this article we have examined the thermalising effect of weak, classical gravitational fields on the Schroedinger wavefunction from the point of view of generalised complex geometry on classical phase space. Using the transformation law for the Schroedinger wavefunction under the passage to an accelerated frame, we have derived the nonrelativistic Unruh effect. As expected, the latter establishes a linear dependence law between the acceleration of the noninertial frame and the temperature thereby generated. Within the scope of the techniques presented here lie other interesting physical systems, to be treated in an upcoming publication. Such are quantum-classical hybrids [36,37] and the thermalising properties of nonuniform (but still weak and classical) gravitational fields.
Altogether, we conclude that generalised complex geometry provides a powerful tool to analyse fluctuation theory and thermal phenomena in the presence of gravity.
Both authors contributed equally to this article. Both authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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