Amit Hagar

hagara@indiana.edu

English

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Length Matters (II): Minimal Length in (Loop) Quantum Gravity and the Fate of Lorentz 
Invariance--A Methodological Dilemma 
 
By: Amit Hagar 
 
In theoretical physics the success of a theory is often achieved when an interesting body of 
mathematical results becomes a plausible explanation of natural phenomena. This process usually 
involves two phases: (1) the theory demonstrates mathematical consistency, and (2) it leads to 
predictions of phenomena that are both new and generic, i.e., new phenomena that are general 
consequences of the mathematical formalism and thus hold fora wide range of parameters as well as for 
generic initial conditions. We usually see the theory as a viable explanation when some of those new 
generic phenonena are observed. 
 
While mathematical consistency is ultimately necessary, it is not sufficient for the success of the theory. 
As the history of science shows, the reasons why theories succeed or fail often have to do with their 
generic consequences. In successful cases the consequences do not stand in conflict with previous 
experiments but are easily confirmed when searched for in new experiments. In unsuccessful cases the 
consequences generically disagree with experiment. Some of these ’bad’ cases still survive for some 
time because the theory has parameters that can be fine–tuned to hide the empirical consequences, but 
ultimately they succumb to lack of predictability (which follows from the same flexibility that allows the 
generic consequences to be hidden). 
 
During the ’phase transition’ from (1) to (2) physicists often come up with heuristic arguments that are 
sufficient to uncover generic consequences of new theories even before precise predictions can be 
stated. For philosophers of physics who defy Otto von Bismarcks famous quip (“Laws are like sausages, it 
is better not to see them being made”), this state of affairs allows an interesting glimpse into the 
practice of theoretical physics. 
 
Such an opportunity presents itself today in the domain of high energy physics, whereat least one 
candidate theory for the unification of quantum mechanics and general relativity — loop quantum 
gravity (LQG) — has reached this transitional stage. In this paper I focus on one of the generic 
consequences of LQG, namely, the fundamental discreteness of space, and on the lively debate on 
whether or not this discreteness entails a departure from exact Lorentz invariance. Since current 
experiments, in the highest energies we can manage, show no sign of departure, LQG theorists are 
facing an interesting methodological dilemma: on one hand they would like to transcend phase (1) and 
rebut the old pessimistic argument that says we would never be able to test quantum gravity effects 
(e.g., Isham 1995); on the other hand they would like to avoid phase (2), as it seems that a generic 
consequence of the theory leads to some fatal results and contradicts experiments. In recent years a 
possible way–out from this dilemma has surfaced (e.g., Amelino Camelia 2002; Smolin 2005), which 
extends the principle of relativity to include a fundamental length scale, and instead of ‘breaking’ 
Lorentz invariance, ‘deforms’ the Lorentz group (hence the name, DSR, for Deformed Special Relativity). 
Among the challenges DSR faces, two are particularly instructive as they best exemplify the 
aforementioned dilemma. The aim of the paper is (i) to expose a common feature of these two 
arguments, and (ii) to draw some general methodological lessons from the similarities they bear toother 
case–studies in theoretical physics. 
 
I begin by surveying the theoretical considerations that have led to the idea of discrete-ness of space in 
theories that aim to unify quantum field theory with gravity (Deser 1957;Wheeler 1957; Mead 1964), 
and proceed by accepting as a working hypothesis the claim that LQG predicts such a discreteness 
(Rovelli & Smolin 1995; Rovelli 2004, pp. 249–259;Rovelli 2007). Next I discuss the intuition that (a) the 
discreteness of space ultimately leads to violations of Lorentz invariance, and the debate that ensues on 
(b) whether such violations indicate violations of the principle of relativity. I complete the setting of the 
stage by briefly presenting DSR which accepts (a) and rejects (b). In the main part of the paper I address 
two heuristic arguments against DSR: the first (Rovelli & Speziale 2003) aims to undermine (a) by arguing 
for a compatibility between discreteness of length and Lorentz invariance. The second (Schutzhold & 
Unruh 2003) aims to expose the price for a negative answer to (b). The main goal of the paper is to 
demonstrate that both these arguments are unconvincing as both assume the universal applicability of 
the principles DSR modifies. Finally, I compare these arguments to other case–studies from the history 
of physics where a similar methodological flaw is present. The analysis is conducive to the broader 
project the author has been engaged in (e.g., Hagar 2007), namely, the extraction of metaphysical 
predilections from the actual practice of theoretical physics. 
 
References: 
[1] Amelino Camelia, G. (2002), Relativity in space–times with short–distance structure governed by an 
observer–independent (Planckian) length scale, Int. J. Mod. Phys.D11:35-60. 
[2] Deser, S. (1957), General relativity and the divergence problem in quantum field theory, Rev.Mod. 
Phys., 29(3): 417-423. 
[3] Hagar, A. (2007), Experimental metaphysics2: The double standard in the information theoretic 
approach to quantum theory, Studies of History and Philosophy of Modern Physics38(4):906–919. 
[4] Isham, C. (1995), Structural issues in quantum gravity, arXiv:gr–qc/9510063. 
[5] Mead, C.A. (1964), Possible connection between Gravitation and fundamental length, Physical 
Review, 135(3B): 849-862. 
[6] Rovelli, C. (2004),Quantum Gravity, Cambridge: Cambridge University press. 
[7] Rovelli, C. (2007), Comment on “Are the spectra of geometrical operators in Loop Quantum Gravity 
really discrete?”, http://xxx.arxiv.org/abs/0708.2481v1. 
[8] Rovelli, C. & Smolin, L. (1995), Discreteness of area and volume in quantum gravity, Nuclear Physics B 
442, pp. 593–619. 
[9] Rovelli, C. & Speziale, S. (2003), Reconcile Planck–scale discreteness and the Lorentz–FitzGerald 
contraction, Physical Review D 67, 064019. 
[10] Schützhold, R. & Unruh, W.G. (2003), Large–scale nonlocality in “Doubly Special Relativity” with an 
energy–dependent speed of light, JETP Letters, 78(7): 431-435. 
[11] Smolin, L. (2005), Falsifiable predictions from semiclassical quantum gravity, Nuclear Physics  B, 742: 
142-157. 
[12] Wheeler, J.A. (1957), On the nature of quantum gemetrodynamics, Annals of Physics, 2: 604-614. 
 


Length Matters (II): Minimal Length in (Loop) Quantum Gravity and the Fate of Lorentz Invariance--A Methodological Dilemma

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Length Matters (II): Minimal Length in (Loop) Quantum Gravity and the Fate of Lorentz Invariance--A Methodological Dilemma

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