Against symmetry Lee Smolin Perimeter Institute for Theoretical Physics

1. Against symmetryLee SmolinPerimeter Institute for Theoretical Physics • Platonism verses Relationalism • What physicists mean by relationalism • General relativity is a (partly) relational theory • Relational space and time in quantum gravity • Relationalism and reductionism • The link to Darwin • Relational quantum theory • Conclusions

2. Two traditions in the search for fundamental physics

3. The Platonic tradition There is another realm in which the laws are revealed in their most perfect form. Symmetry= perfection Lack of symmetry=decay=distance from the perfect realm

4. The Platonic tradition There is another realm in which the laws are revealed in their most perfect form. Symmetry= perfection Lack of symmetry=decay=distance from the perfect realm A symmetry is a substitution of one entity for another that does not change any physical property of either.

5. The Platonic tradition There is another realm in which the laws are revealed in their most perfect form. Symmetry= perfection Lack of symmetry=decay=distance from the perfect realm In its modern version: The perfect realm is found in the limit of short distance of high energy The imperfect realm of our experience is the result of symmetry breaking A more unified theory is one that has more symmetry Elementary particles are classified by how they transform under the symmetry. The ground state has as much symmetry as possible. There is a spacetime background which is characterized by maximal symmetry.

6. The Leibnizian tradition: The principle of identity of the indescrernible implies that there are no two distinct entities with the same properties. Hence when the laws of nature are expressed in terms of fundamental entities there can be no symmetries. This implies: There are no absolute properties defined with respect to an eternally unchanging background. Instead, each existing entity has relational properties, given by their relation to all the rest.

7. Leibnizian Platonic Space: absolute, eternal emergent, relational, background evolving Properties: defined with respect all defined relationally to the background Dynamics: background background dependent independent Symmetries: maximal none Classical Newtonian general relativity theory dynamics Present string theory loop quantum gravity, Incarnation: causal sets, dynamical triangulations, NCG etc.

8. The search for a complete theory by the background dependent route. This has been the main pre-occupation of theoretical physicists the last 40 years. It is now pretty clear that we have failed. Why we have failed is a question that may be of interest for philosophers. The big ideas have been: 1) unification. All elementary particles and forces arise from different states or solutions of one elementary entity. 2) symmetry: Unification is to be achieved by increasing the symmetry of physical law (for example by increasing the dimension.) Our world then corresponds to one particularly asymmetric solution to symmetric physical laws. First try: unified field theory. Einstein, Weyl, Kaluza, Klein, 1920-1955 Second try: string theory. 1984-2005

9. The big hope:There exists exactly one consistent unified theory of all the interactions and particles. The present evidence: There is evidence that string theories (on fixed backgrounds) are consistent to 2nd order in a certain approximation scheme. There is evidence that there are at least 10500 distinct string theories, which at least to low order provide unifications of gauge fields, gravity and fermions, consistent with positive cosmological constant. However, of the much smaller number that are understood in any detail, all make predictions that disagree with observations. It has been conjectured that all string theories are unified in one big background independent theory M theory. But, in spite of much effort by many smart people, this theory has never been written down.

10. Another big hope:The more things are unified, the most symmetry the theory has, and the more unique it should be and the fewer free parameters it should have. The results to date: The standard model of elementary particle physics: 20 free parameters Its simplest supersymmetric extension: 125 free parameters The number of distinct string theories that could reproduce them: 10500 So it appears that the more symmetry and the more unification, the more free parameters and the less uniqueness. WHY?

11. The background dependent approach has one more move to play: The anthropic hope: There are a vast number of unified theories, and a vast number of regions of the universe where they may act. Out of all of these, there will be a very small fraction where the laws of physics allow the existence of intelligent life. We find ourselves in one of these. Because the number of universes and theories is so vast, theory can make no prediction except those that follow from requiring our own existence. (Susskind, Schwartz, Douglas, Linde, Polchinski and others...) The result is a reducto ad absurdum of taking the background dependent approach, in which, recently, good people have found no alternative but to seriously argue for the adequacy of theories that make no falsifiable predictions. Can we do better by going back and taking the relational approach more seriously?

12. My main theme: We should believe relationalism, in the sense I will define it, not just because there are good philosophical arguments for it (Leibniz, Mach, etc...) but because it leads to theories that are more tightly constrained, and hence more falsifiable. I will describe two (3, if time) contemporary examples of this: 1) Quantum gravity 2) The search for a complete theory of nature. 3) Is there a quantum theory for systems like the universe that contain their own observers?

13. My main theme: We should believe relationalism, in the sense I will define it, not just because there are good philosophical arguments for it (Leibniz, Mach, etc...) but because it leads to theories that are more tightly constrained, and hence more falsifiable. I will describe two (3, if time) contemporary examples of this: 1) Quantum gravity 2) The search for a complete theory of nature. 3) Is there a quantum theory for systems like the universe that contain their own observers? But first, we need a more precise definition of relationalism.

14. What do physicists mean by relationalism? • The world is made of a large number of entities or events • How do they get their properties?

15. What do physicists mean by relationalism? • The world is made of a large number of entities or events • How do they get their properties? • In an absolute scenario, there is an external and static entity, such as Newton’s absolute space, and properties of elementary particles are defined individually in terms of their relations to the absolute entity. • Hence, a particle in Newton’s absolute space has the same properties whether it is one of many or the only thing in the universe. • The absolute entities make up the background.

16. What do physicists mean by relationalism? • The world is made of a large number of entities or events • How do they get their properties? • In an absolute scenario, there is an external and static entity, such as Newton’s absolute space, and properties of elementary particles are defined individually in terms of their relations to the absolute entity. • Hence, a particle in Newton’s absolute space has the same properties whether it is one of many or the only thing in the universe. • The absolute entities make up the background. • The most basic statement of relationalism is: • R1 There is no background

17. How then do we understand the properties of elementary particles? The relational view posits that R2 The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities.

18. How then do we understand the properties of elementary particles? The relational view posits that R2 The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities. Examples of purely relational systems: Graph

19. How then do we understand the properties of elementary particles? The relational view posits that R2 The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities. Examples of purely relational systems: Graph Partially ordered set

20. How then do we understand the properties of elementary particles? The relational view posits that R2 The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities. Examples of purely relational systems: Graph Partially ordered set Example of a partly relational system: knots and links

21. What is time in a relational theory? R3 The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering. Relationalism is also a research strategy: Relational strategy: Seek to make progress by identifying the background structures in our theories and removing them, replacing them with relations between physical entities which evolve subject to dynamical law.

22. Summary: relationalism according to physicists: R1 There is no background. R2 The fundamental properties of the elementary entities consist entirely in relationships between those entities. R3 The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering. Relational strategy: Seek to make progress by identifying the background structures in our theories and removing them, replacing them with relations which evolve subject to dynamical law. absolute vrs relational background dependent vrs background independennt

23. General relativity is a partly relational theory: Layers of structure:Dimension Topology M Differential structure Metric and fields gab, f In GR: Mis fixed. gab and f describe relational information. KEY POINT: A physical spacetime is NOT modeled by a manifold, metric, and fields, but by an equivalence class of manifolds and metrics, which are equivalent under any diffeomorphism !! A diffeomorphism is a smooth map from M to itself that takes differential functions to differential functions. f q p

24. What information is coded inside an • equivalence class? • Not fields at points. because physical points are • only identified by what happens there. • The causal structure. i.e. which events are causally related to which? • The measure. i.e. what is the volume of each set defined by the causal structure? • It can be shown that the information in a spacetime • {M, gab, f } is completely characterized by the causal structure and the measure. • Hence, once the dimension, topology and diff structure are fixed, • the physical content of GR is about the causal relations among physical events. f q p

25. 3. Relational space and time in quantum gravity • Conventional quantum mechanics and quantum field theory • (QFT) are background dependent theories. • Background structures: background geometry, inner product, • external clocks and observers • Hence there are two options: • Background dependent: give up relationalism, and quantize • gravitational waves on fixed backgrounds, using conventional • QFT methods. • perturbative quantum GR, string theory... • Background independent: Find a way to define a quantum • theory which is relational and applies to relational theories. • causal sets, loop quantum gravity, dynamical triangulations

26. A purely relational approach to quantum spacetime: causal sets • Def: A causal set is a finite partial order • a --> b means a causally precedes b. • Postulates: • A quantum spacetime history, h, is nothing but a • causal set. • The dynamics is given by a rule that assigns a quantum • amplitude A[h], a complex number, to each history. • The amplitude to go from initial to final state is given by the • sum over amplitudes for each history that does so.

27. The motivation for the causal set approach: • The physical information in GR (apart from M) is coded in the causal relations amongst events. • But quantum spacetimes should be discrete. • A classical spacetime can be approximated by a causal set. • Def: A causal set C approximates a spacetime {M,gab} when • there is an embedding C --> {M,gab} that preserves the causal structure, with one event in the image per Planck volume. • But there is a problem:

28. The inverse problem: Every classical spacetime {M,gab} can be approximated by a causal set, and in many ways.

29. The inverse problem: Every classical spacetime {M,gab} can be approximated by a causal set, and in many ways. But for almost no causal set C is there a spacetime {M,gab} of low dimension, that it approximates. ?

30. This is an example of a more general inverse problem facing any discrete approach to quantum gravity: Its easy to approximate smooth fields with discrete structures.

31. Its easy to approximate smooth fields with combinatoric structures. But generic graphs do not embed in manifolds of low dimension, preserving even approximate distances. ? Those that do satisfy constraints unnatural in the discrete context,

32. Loop quantum gravity: provides a robust method for the • quantization of diffeomorphism invariant theories. • Consider a classical gravitational theory, T, whose histories are • described as diffeomorphism equivalence class of metrics and fields, • {(M, gab, f), }. • Assume that: • The topology and dimension of spacelike surfaces, , are fixed. • The metric and fields have dynamics given by the Einstein equations or some natural extension. Note that the dynamics can be expressed in an equivalent formulation in which the configuration space is that of a gauge field Aa on a spatial manifold and the metric information is in the electric field.

33. The fundamental theorem: Consider a background independent gauge theory, compact Lie group G on a spatial manifold S of dim >1. No metric!! (G=SU(2) for 3+1 gravity) There is a unique cyclic representation of the algebra generated by Wilson loops and electric flux in which the Hilbert space carries a unitary rep of the diffeomorphism group. Lewandowski, Okolo, Sahlmann, Thiemann+ Fleishhack(LOST theorem) This means that there is a unique diffeomorphism invariant quantum quantum theory for each G. Ashtekar: GR is a diffeomorphism invariant gauge theory!! Hence this class of theories includes loop quantum gravity and spin foam models

34. This gives rise to causal spin network theories • Pick an algebra G • Def: G-spin network is a graph G with edges labeled by representations of G and vertices labeled by invariants. • Pick a differential manifold S. • {G } an embedding of G in S, up to diffeomorphisms • Define a Hilbert space H: • |{G }> Orthonormal basis element for each {G } • Define a set of local moves and give each an amplitude • A history is a sequence of moves from an in state to an out state • Each history has a causal structure

35. Basic results of loop quantum gravity: 1) The quantum spacetime is discrete in that each node of the graph corresponds to a finite quanta of spatial volume. The operators that correspond to volumes, areas and lengths are finite, and have discrete spectra with finite non-zero minimal values. Hence a graph with a finite number of nodes and edges defines a region of space with finite volume and area.

36. 3) Any history of the quantum theory is defined by a series of local moves on graphs that take the initial state to the final state. The set of local moves in each history define a causal set. 4) The amplitudes for local moves that follow from the quantization of the Einstein equations are known in closed form. The sums over those amplitudes are known to be ultraviolet finite.

37. 5) Similarly, the quantum Einstein equations in the Hamiltonian form have been implemented by exact operator equations on the states. Many exact solutions are known. • The entropy of black holes is understood exactly in terms of the quantum geometry of horizons.

38. It has recently been proven that cosmological singularities bounce, so the evolution of the universe continues through apparent classical cosmological singularities. This has led to a prediction for an effect, observable with current CMB data. 8) These theories have generically excitations whose coherence is maintained by topological conservation laws. These are candidates for elementary particles. 9) There are known explicit candidate ground states, whose excitations reproduce the physics of quantum fields and linearized gravitational waves on fixed backgrounds.

39. Open problems of loop quantum gravity: • Classical limit:Find the quantum state which is in fact • the ground state of the theory and show that it reproduces • quantum field theory and classical GR from its excitations. • Do science: Make predictions for doable experiments • that can test the theory up or down. • Remove the remaining background dependence: The results • so far defined depend on the fact that the dimension and • topology,  of the spatial manifold is fixed, so that the graphs are • embedded in . This helps by lessoning the inverse problem. • Can this be removed-and the inverse problem solved-so that all • the structure that was background for previous theories, including • dimension and topology, is explained as following from solutions • to a relational theory?

40. The most important news:It is possible to probe the Planck scale experimentally, to search for genuine quantum gravity effects. The reason:The discrete structure of space and time leads to corrections to energy momentum relations and (possibly) to the actions of the Loretz transformations at short distances: E2 = p2 + m2 + a lp E3 +… These are observable, and a is expected to be measured or bounded in experiments now in progress: AUGER: ultra high energy cosmic rays GLAST: Gamma ray busts (because of a resulting energy dependence of the speed of photons.) Loop quantum gravity may provide robust predictions for these effects. It does in 2+1 dimensions, and there are recent calculations that give predictions for them in 3+1 dimensions, that depend on the choice of ground state.

41. Lessons for the absolute/relational debate: • Loop quantum gravity is the most successful background • independent quantum theory of gravity. It works by being • conservative and sticking to the principles of GR and QM. • Thus LQG is partly relational, in exactly the same way that GR • is partly relational: the dimension, topology and matter fields are • fixed, the metric of spacetime is described quantum mechanically • in purely relational terms, in terms of evolving labeled graphs • embedded in the spatial topology. • The main barrier to making an entirely relational theory of • quantum spacetime appears to be the inverse problem. • But even so, loop quantum gravity, as a partly relational theory is • more tightly constrained, and more testable than non-relational • alternatives.

42. Reductionism vrs relationalism Common sense reductionism: When a system has parts, it makes sense to base an understanding of it on the laws that the parts satisfy, as well as on patterns that emerge from the exchanges of energy and information among the parts. But this has a built in limit: What do we do when we get to elementary entities that have no parts? How do we explain the properties of the elementary particles? Two options: In a background dependent theory: the properties of the elementary particles are given by their relations to the background. In a background independent theory: the properties of the elementary particles can arise only from their relations to each other.

43. A relational complete theory must proceed differently. Its guiding principle must be asymmetry rather than symmetry. WHY? Leibniz’s principle of the identity of the indiscernible: If two entities have the same relations to the rest, they are to be identified. Each individual entity must then have a unique set of relations to the rest. The fundamental entities in spacetime are the events. The relation of an event to the others is coded in the information that arrives to it from the past. We may call this the view of the event. Spacetime must be so asymmetric that there are no two events with identical views. The universe cannot have any symmetries The universe cannot be in thermal equilibrium

44. Hence, a relational complete theory must have mechanisms that drive the universe away from symmetry and equilibrium. Interestingly, gravity does both these things. (This is interesting because by Einstein gravity is the force that exists because space and time are relational.) Are there others?

45. Hence, a relational complete theory must have mechanisms that drive the universe away from symmetry and equilibrium. Interestingly, gravity does both these things. (This is interesting because by Einstein gravity is the force that exists because space and time are relational.) Are there others? Natural selection.

46. Hence, a relational complete theory must have mechanisms that drive • the universe away from symmetry and equilibrium. • Natural selection is in some senses relational theory: • The properties natural selection acts on, such as fitness, are relational quantities, they are meaningless for a world with a single species. • Natural selection follows the relational strategy. Properties such as species were believed to be eternal, and a priori become relational and historical. • Could natural selection or something like it account for the • choice of physical laws and their parameters? • Could it account for the anthropic observation? Our universe is • much more complex (in for example its astrophysics and chemistry) • than most universes with the same laws but different values of the • parameters of those laws (including masses, charges, etc.)

47. To apply natural selection to a system it must have: • A space of parameters for each entity, such as the genes. • A mechanism of reproduction. • A mechanism for those parameters to change, but slightly, • from parent to child. • Reproductive success depends strongly on the parameters. • This can be applied to cosmology: • The space of parameters is the space of parameters of the standard • models of physics and cosmology. This leads to the term “the string • theory landscape.” • The mechanism of reproduction is the formation of black holes. • We may conjecture that the low energy parameters do change in • such a bounce. • The mechanism of differentiation is that universes with • different parameters will have different numbers of black holes.

48. These lead to genuine falsifiable physical predictions: • Most ways to change the parameters of the standard models of particle • physics and cosmology should have fewer black holes. • There can be no neutron stars with masses larger than 1.6 times the • mass of the sun.

49. In these instances, the relational theory turns out to be more predictive, and more falsifiable than background dependent theories. WHY? The difference is between: 1) Explanations that refer ultimately to a network of relationships amongst equally physical entities, which evolve dynamically. vrs 2) Explanations that refer to relationships between dynamical entities and an a priori, non-dynamical, background. The former are more constrained, hence harder to construct. More of what is observed is subject to law, as there is no background to be freely chosen. Hence, relational, background independent theories are more testable, and more explanatory.

50. Relationalism and quantum theory