Loop Quantum Brans-Dicke Theory
The loop quantization of Brans-Dicke theory (with coupling parameter ) is studied. In the geometry-dynamical formalism, the canonical structure and constraint algebra of this theory are similar to those of general relativity coupled with a scalar field. The connection dynamical formalism of the Brans-Dicke theory with real -connections as configuration variables is obtained by canonical transformations. The quantum kinematical Hilbert space of Brans-Dicke theory is constituted as of that loop quantum gravity coupled with a polymer-like scalar field. The Hamiltonian constraint is promoted as a well defined operator to represent quantum dynamics. This formalism enable us to extend the scheme of non-perturbative loop quantum gravity to the Brans-Dicke theory.
Department of Physics, Beijing Normal University, Beijing 100875, China
E-mail: email@example.com; firstname.lastname@example.org
In the past 25 years, loop quantum gravity(LQG), a background independent approach to quantize general relativity (GR), has been widely investigated [1, 2, 3, 4]. Recently, this non-perturbatively loop quantization procedure has been generalized to the metric theories[5, 6]. In fact, modified gravity theories have recently received increased attention in issues related to ”dark Universe” and non-trivial tests on gravity beyond GR. Besides theories, a well-known competing relativistic theory of gravity was proposed by Brans and Dicke in 1961 , which is apparently compatible with Mach’s principle. To represent a varying ”gravitational constant”, a scalar field is non-minimally coupled to the metric in Brans-Dicke theories(BDT). On the other hand, since 1998, a series of independent observations implied that our universe is currently undergoing a period of accelerated expansion. These results have caused the ”dark energy” problem in the framework of GR. It is reasonable to consider the possibility that GR is not a valid theory of gravity on a galactic or cosmological scale. The scalar field in BDT of gravity is then expected to account for ”dark energy”. Furthermore, a large part of the non-trivial tests on gravity theory is related to Einstein’s equivalence principle (EEP) . There exist many local experiments in solar-system supporting EEP, which implies the metric theories of gravity. Actually, BDT are a class of representative metric theories, which have been received most attention. Thus it is interesting to see whether this class of metric theories of gravity could be quantized nonperturbatively. Note that the metric theories are equivalent to the special kind of BDT with the coupling parameter and some non-vanishing potential of the scalar field. In this work, for simplicity consideration, we only consider BDT with coupling parameter . The connection formalism of BDT is derived from its geometrical dynamics. Based on the resulted connection dynamical formalism, we then quantize the BDT by extending the nonperturbative quantization procedure of LQG in the way similar to loop quantum gravity. Throughout the paper, we use Greek alphabet for spacetime indices, Latin alphabet a,b,c,…, for spatial indices, and i,j,k,…, for internal indices.
2 Classical and Quantum Aspects of Brans-Dicke Theories
The original action of Brans-Dicke theories reads
where we set , denotes the scalar curvature of spacetime metric , The Hamiltonian analysis of BDT can be found in Refs.[11, 12]. By doing 3+1 decomposition of the spacetime, the four-dimensional scalar curvature can be expressed as
where is the extrinsic curvature of a spatial hypersurface , , denotes the scalar curvature of the 3-metric induced on , is the unit normal of and is the lapse function. By Legendre transformation, the momenta conjugate to the dynamical variables and are defined respectively as
where is the shift vector. The resulted Hamiltonian of BDT can be derived as a liner combination of constraints as where the smeared diffeomorphism and Hamiltonian constraints read respectively
Here the condition was assumed. Lengthy but straightforward calculations show that the constraints comprise a first-class system similar to GR. Since the geometric canonical variables of BDT are as same as those of theories , we can use the same canonical transformations of theories to obtain the connection dynamical formalism of BDT. Let
The new geometric variables are and where is the triad such that , , is the spin connection determined by , and is a nonzero real number. It is clear that our new variable coincides with the Ashtekar-Barbero connection [13, 14] when . The only non-zero Poisson brackets among the new variables reads Now, the phase space of BDT consists of conjugate pairs and , with the additional Gaussian constraint which justifies as an -connection. The original vector and Hamiltonian constraints can be respectively written up to Gaussian constraint as
where is the curvature of . All the constraints are of first class. The total Hamiltonian can be expressed as a linear combination
Based on the connection dynamical formalism, the nonperturbative loop quantization procedure can be straightforwardly extended to the BDT. The kinematical structure of BDT is as same as that of theories [5, 6]. The kinematical Hilbert space of the system is a direct product of the Hilbert space of geometry and that of scalar field, , with the orthonormal spin-scalar-network basis over some graph . Here and consist of finite number of curves and points respectively in . The basic operators are the quantum analogue of holonomies of connections, densitized triads smeared over 2-surfaces , point holonomis , and scalar momenta smeared on 3-dimensional regions . Note that the whole construction is background independent, and the spatial geometric operators of LQG, such as the area , the volume  and the length operators [18, 19] are still valid here. As in LQG, it is straightforward to promote the Gaussian constraint to a well-defined operator[2, 4]. It’s kernel is the internal gauge invariant Hilbert space with gauge invariant spin-scalar-network basis. Since the diffeomorphisms of act covariantly on the cylindrical functions in , the so-called group averaging technique can be employed to solve the diffeomorphism constraint[3, 4]. Thus we can also obtain the desired diffeomorphism and gauge invariant Hilbert space for the BDT.
Now, we come to implement the Hamiltonian constraint (9) at quantum level. In order to compare the Hamiltonian constraint of BDT with that of theories in connection formalism, we write Eq. (9) as . It is easy to see that the terms just keep the same form as those in theories, the terms are also similar to the corresponding terms in theories. Here differences are only reflected by the coefficients. Now we come to the completely new term, . We can introduce well-defined operators as in Ref. . By the same regularization techniques as in Refs.[6, 20], we triangulate in adaptation to some graph underling a cylindrical function in and reexpress connections by holonomies. The corresponding regulated operator can acts on a basis vector over some graph . It is easy to see that the action of on is graph changing. It adds a finite number of vertices at for edges starting from each high-valent vertex of . As a result, the family of operators fails to be weakly convergent when . However, due to the diffeomorphism covariant properties of the triangulation, the limit operator can be well defined via the so-called uniform Rovelli-Smolin topology induced by diffeomorphism-invariant states . It is obviously that the limit is independent of . Hence the regulators can be removed. We then have
Thus the total Hamiltonian constraint operator is well defined in . Furthermore, master constraint programme can be introduced for BDT to avoid possible quantum anomaly and find the physical Hilbert space.
With the key observation that LQG is based on its -connection dynamical formalism which can be derived via canonical transformations from the geometric dynamics, the -connection dynamics of BDT is obtained. Thus LQG has been successfully extended to the BDT by coupling to a polymer-like scalar field. The quantum kinematical structure of BDT is as same as that of loop quantum theories. Hence the important physical result that both the area and the volume are discrete remains valid for quantum BDT. While the dynamics of BDT is more general than that of theories, the Hamiltonian constraint can still be promoted to a well-defined operator in . Hence the classical BDT can be non-perturbatively quantized. Therefore, besides GR and theories, LQG method is also valid for the BDT of gravity.
We thank the organizers of Loops 11 conference for the financial support to our attendance. This work is supported by NSFC (Grant No.10975017) and the Fundamental Research Funds for the Central Universities.
-  C. Rovelli, Quantum Gravity, (Cambridge University Press, 2004).
-  T. Thiemann, Modern Canonical Quantum General Relativity, (Cambridge University Press, 2007).
-  A. Ashtekar and J. Lewandowski 2004 Class. Quant. Grav. 21 R53
-  M. Han, Y. Ma and W. Huang 2007 Int. J. Mod. Phys. D 16 1397
-  X. Zhang and Y. Ma 2011 Phys. Rev. Lett. 106 171301
-  X. Zhang and Y. Ma 2011 Phys. Rev. D 84 064040
-  C. Brans and R. H. Dicke 1961 Phys. Rev. 124 925
-  J. Friemann, M. Turner, D. Huterer 2008 Ann. Rev. Astron. Astrophys. 46 385
-  C. M. Will 2006 Living Rev. Rel. 9 3
-  T. P. Sotiriou and V. Faraoni 2010 Rev. Mod. Phys. 82 451
-  X. Zhang and Y. Ma, arxiv:1107.5157.
-  G. J. Olmo and H. Sanchis-Alepuz 2011 Phys. Rev. D 83 104036
-  A. Ashtekar 1986 Phys. Rev. Lett. 57 2244
-  J. Barbero 1995 Phys. Rev. D 51 5507
-  A. Ashtekar, J. Lewandowski, H. Sahlmann 2003 Class. Quant. Grav. 20 L11
-  C. Rovelli and L. Smolin 1995 Nucl. Phys. B 442 593
-  A. Ashtekar and J. Lewandowski 1998 Adv. Theor. Math. Phys. 1 388
-  T. Thiemann 1998 J. Math. Phys. 39 3372
-  Y. Ma, C. Soo, J. Yang 2010 Phys. Rev. D 81 124026
-  M. Han and Y. Ma 2006 Class. Quant. Grav. 23 2741