An Adynamical, Graphical Approach to Quantum Gravity and its Foundational
Implications
Michael Silberstein
Elizabethtown College
UMD, College Park, Foundations of Physics Group
Mark Stuckey
Elizabethtown College
Introduction. We propose a path integral over graphs approach to quantum gravity and
unification that requires a modification and reinterpretation of both general relativity (GR) and
quantum field theory (QFT) via their graphical instantiations, Regge calculus and lattice gauge
theory (LGT), respectively. As we outline below, the spacetime metric and the matter and gauge
field gradients on the graph are co-determining, so there is no “background spacetime” connoting
existence independent of matter-energy-momentum, and the graphical action can be
characterized geometrically via graphical boundary operators (Silberstein et al, 2013).
Our graphical approach possesses “disordered locality,” i.e., links can be arbitrarily long, similar
to that of quantum graphity (Caravelli & Markopoulou, 2012; Prescod-Weinstein & Smolin,
2009) and, as is the case with quantum graphity, this is used to explain dark energy (more on that
below). However, our approach differs from all others in that it is adynamical (Silberstein et al,
2012). Carroll sums up nicely what we mean by a dynamical approach (Carroll, 2012):
Let’s talk about the actual way physics works, as we understand it. Ever since Newton,
the paradigm for fundamental physics has been the same, and includes three pieces. First,
there is the “space of states”: basically, a list of all the possible configurations the
universe could conceivably be in. Second, there is some particular state representing the
universe at some time, typically taken to be the present. Third, there is some rule for
saying how the universe evolves with time. You give me the universe now, the laws of
physics say what it will become in the future. This way of thinking is just as true for
quantum mechanics or general relativity or quantum field theory as it was for Newtonian
mechanics or Maxwell’s electrodynamics.
Carroll goes on to say that all extant formal models of quantum gravity, even those attempting to
recover spacetime, are dynamical in this sense. This means that in the entire history of physics,
even among those thinkers who defend eternalism or blockworld, based on relativity or what
have you, have at least methodologically and formally assumed dynamism. While it is true that
integral calculus and least action principles have been around for a long time, most assume these
methods are formal tricks and not fundamental to dynamical equations. While our adynamical
approach requires mathematical formalism as daunting as our dynamical competitors, our
formalism seeks to redefine what it means to “explain” something in physics. Rather than finding
a rule for time-evolved entities per Carroll, we seek a rule for the self-consistent co-construction
of spacetime metric and matter fields on the graph. One might ask for example, “Why does link
X have metric Y and stress-energy tensor Z?” A dynamical answer might be, “Because link X-1
has metric Y-1 and stress-energy tensor Z-1 and the law of evolution thereby dictates that link X
has metric Y and stress-energy tensor Z.” Notice how this answer is independent of future
boundary conditions; indeed, it’s independent of conditions anywhere else on the graph other
than those of the immediate 3D hypersurface. Contrast this with an adynamical answer such as,
“Because the values Y and Z on X satisfy the global self-consistency criterion for the graph as a
whole.” We believe it reasonable to seek such a global “self-consistency criterion” (SCC) in the
path integral formalism where action is extremized in 4D.
At this time we have only an idea for the SCC (outlined below), but as with quantum graphity,
the idea itself has proven useful for explaining dark energy via disordered locality. Not
surprisingly, the two explanations, both graphical and both dependent on a particular form of
quantum gravity, deviate greatly in their conclusions. The dynamical approach of quantum
graphity generates a cosmological constant (dark energy) to generate accelerated expansion as in
ɅCDM, i.e., Einstein-deSitter cosmology (EdS) plus a cosmological constant Ʌ. Our adynamical
approach modifies cosmic distance in the distance modulus of EdS and the resulting distance
modulus versus redshift fit of the Union2 supernova data is equivalent to that of ɅCDM
(Stuckey et al, 2012a). Thus, in contrast to quantum graphity, we have decelerating expansion
and no need for dark energy (Stuckey et al, 2012b). We suspect similar analysis of the graphical
Schwarzschild solution will negate the need for dark matter.
We will explain foundational implications of this adynamical approach for quantum mechanics
(Silberstein et al, 2008 & 2013; Stuckey et al, 2008), quantum field theory (Stuckey et al, 2013)
and general relativity. For example with regard to quantum physics, our interpretation is a form
of relational fundamentalism, i.e., discrete graphs at bottom, and is based on an actual formal
model that does potentially advance physics—it is not merely interpretation of existing
formalisms. Our model can be thought of as a kind of psi-epistemic hidden variable account with
the graphical structure and the SCC as the hidden variable. Unlike most psi-epistemic accounts
such as quantum Bayesianism, ours has an underlying physical explanation as to why psi has the
epistemic status; there is no hidden instrumentalism. The story is, if you construct the differential
equation corresponding to the path integral Z (transition amplitude), the time-dependent foliation
is the wavefunction ψ(t), which becomes of interest only when you don’t know when the
outcome is going to occur. Once you have your outcome, ψ = Z is fixed and the relevant/related
distribution of spacetime regions is established. Quantum physics then is simply providing a 4D
distribution function for graphical relations responsible for the experimental equipment and
process from initiation to termination. On our view a dynamical 3D-object-activity-oriented
ontological bias is responsible for confusion concerning the time-evolved wavefunction ψ(t) of
Schrodinger dynamics. The wavefunction it turns out is not an instant by instant updating of the
probabilities of outcomes caused by “quantum objects” interacting with the objects of the
experiment (beam splitters, mirrors, detectors, etc). From the graphical perspective at bottom
there is no separation between the so-called quantum system and the experimental apparatus. As
for quantum entanglement or non-locality, i.e., “disordered locality,” it is ultimately explained at
the graphical level via a violation of statistical independence between physical properties at
distinct points in spacetime. Our account of the quantum is certainly time-symmetric but requires
no backwardly causal or time-reversed processes. Rather, quantum phenomenon is explained in
terms of acausal/adyamical global constraints fundamental to the quantum itself. There is much
more to be said about the specific interpretative conundrums of QFT. (Stuckey et al, 2013).
Technical Overview (details can be found in Stuckey et al, 2013). Rather than particles and
forces as standardly understood in the algebraic continuum approaches to QFT, we propose an
alternative account of fields a la LGT, wherein one uses a path integral of field configurations on
a graph to approximate QFT. In our view, the spacetime metric and the matter and gauge field
gradients on the graph of LGT are co-determining, so there is no “background spacetime”
connoting existence independent of matter-energy-momentum, and the graphical action can be
characterized geometrically via graphical boundary operators. Further, we assume the graphical
approach is fundamental, not the continuum approach of QFT.
To illustrate this idea, consider the Lagrangian density L for the propagation of photons
1
L
F  F
4 o
The field strength tensor (aka curvature) on the graph of Figure 1 is given by
  A x   A0 0  A1 t   A1 0 E x
F01    0

 c
x
ct


1
1
F01 2  1  o E x2  u where u is the energy density of the
F  F 
so that L  
4 o
2 o
2
electric field representing the exchange of a photon of energy E via a field gradient (Figure 1).
To illustrate the relationship between momentum exchange and field gradients, we look to
1
F01 2  xyzt  o E x2  xyztu  Et ,
quantum interference. The graphical action is S [ A]  xyzt
2 o
2
so the probability amplitude (two point correlation function) is proportional to
e
i
S [ A]

e
i
Et

e
i
hct

e
i
2x

Thus, we see that momentum transfer (field gradient between links) p 
h
is related to the

metric per λ, as measured via an interference pattern. Therefore, we refer to λ as the interference
length.
Of course, this is also the case for matter exchange (field gradient between nodes), per
the DeBroglie wavelength λ. To model the exchange of mass, we use LGT for the Klein-Gordon
(KG) equation (whence the Schrödinger equation in the non-relativistic limit). The amplitude in
this case goes as
i
e
Using E 
p 2 c 2  m 2 c 4 and
2
px Et



e
i
2x Et



mc 1
 , where ℓ is a ‘range’ (ℓ = ∞ for photons), we have
h

2
 1  1
E  hc      . Thus, the correlation length (exponential temporal damping of the
  
probability amplitude) per E is a function of λ and ℓ, i.e., energy, momentum and mass
exchange, given by matter and gauge field gradients, can be characterized by the metric and
vice-versa.
Finally, we provide a geometric basis for the graphical action via graphical boundary operators.
It is in such a geometric basis that we seek the SCC. The gauge field contribution to the action on
the graph of Figure 1 is given by the square of the curvature (aka field strength)

F 01 F01  A 0 0 A 0 x  A1 0
 1 
x

1


A1 t   x   1
x
1
 ct 
 1 
 ct 


1
x
 Ao 0
 A x 
1
 1  o   ( 2 A) † ( 2 A)
ct
ct  A1 0


 A1 t  

where  2 is the graphical boundary operator describing the construct of plaquettes from links.
Thus, L  
1
4 o
F  F  
1
2 o
( 2 A) † ( 2 A) . Similarly, we can construct the graphical action
for the KG equation using 1 , the graphical boundary operator describing the construct of links
†
1
1
( 1Q) † ( 1Q)  mQ mQ . To generalize this to vector fields on the
2
2
nodes, we understand that the KG operator for real scalar fields is the square of the operator for
from nodes, i.e., L 



 

complex vector fields ψ, i.e., i     m i     m   2  m 2 , whence the Dirac operator
used for the fundamental fermion fields of the Standard Model. In the graphical action, we are
now expected to take differences between vectors at two different nodes and that requires a
means of parallel transport. So, in our view and that of LGT, local gauge invariance is seen as a
modification to the matter field gradient on the graph required by parallel transport per Uµ, i.e.,
 U 0~0    1  U 1~1   
 D   
 
  ...
ct
x




~
where  i is the complex vector field on the node adjacent to  in the positive ith direction. The


0

1
1
F  F is therefore seen as the addition of
Lagrangian density L  ~   D  m  
2
4 o

†

parallel transport Uµ and a curvature term A †  2  2 A , where A generates Uµ, to
1
†
L     1  1  m 2  to produce a well-defined field gradient between ~i and  .

2 
The graphical view presented here of the Standard Model is just that of LGT, so what are we
proposing differently? We are proposing the graphical view of LGT is fundamental rather than
its continuum limit (QFT). Since we are not requiring one take the limit as link lengths go to
zero, as in LGT, we refer to our approach as modified LGT (MLGT). In this view, unification is
about connecting MLGT to GR in adynamical fashion via the SCC.
Along these lines, we are seeking a mathematical nexus from MLGT to a modified version of
GR’s graphical instantiation, Regge calculus. In Regge calculus, the spacetime manifold is
replaced by a lattice geometry where each 4D cell (simplex) is Minkowskian (flat). Curvature is
represented by “deficit angles” (Figure 2) about any plane orthogonal to a “hinge” (triangular
side to a tetrahedron, which is a 3D side of a 4D simplex). The Hilbert action for a 4D vacuum
1
lattice is I R 
  i Ai where σi is a triangular hinge in the lattice L, Ai is the area of σi and εi
8  i L
is the deficit angle associated with σi. The counterpart to Einstein’s equations is then obtained by
I
demanding R2  0 , where ℓj2 is the squared length of the jth lattice edge , i.e., the metric. To
 j
obtain equations in the presence of matter-energy, one simply adds the appropriate term IM-E to IR
I
I
and carries out the variation as before to obtain R2   M 2 E . One finds the stress-energy
 j
 j
tensor is associated with lattice edges, just as the metric, and Regge’s equations are to be
satisfied for any particular choice of the two tensors on the lattice. Since classical field equations
are given by the extremum of probability amplitudes with respect to the matter and gauge fields,
we believe it may be possible to produce Regge’s equations from variation of our MLGT
probability amplitudes with respect to the metric, as found in the graphical boundary operators of
the action. As with MLGT, we hold Regge’s equations fundamental to GR, so we are connecting
MLGT to modified Regge calculus (MORC). The equations of MORC would then result from
demanding the same solution on the tetrahedron shared by two neighboring simplices. Ordinary
Regge calculus would obtain from MORC by removing disordered locality and allowing for
vacuum simplices. Thus, in our approach (MORC), it becomes clear as to why the Union2 data
was (mistakenly) believed to entail accelerating expansion.
Figure 1
A1(t)
2
A 0(0)
4
A 0(x)
1
3
A 1(0)
Figure 2
Reproduced from Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San
Francisco (1973), p. 1168.
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