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Locality, Quantum Gravity and Metastrings

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Locality, Quantum Gravity and Metastrings Locality, Quantum Gravity and Metastrings Rob Leigh University of Illinois at Urbana-Champaign based on a series of papers, past and future, with Laurent Freidel (Perimeter) and Djordje Minic (VaTech) (see 1502.08005 for details) August 12, 2015 CERN-CKC TH Institute on Duality Symmetries in String and M-theories Metastrings - in this talk, I want to introduce you to the concept of metastrings - as you will see, it is topical to this workshop, as some of its features are reminiscent of double field theory, and concepts such as T-duality are central • however, this is meant to be a full string theory, not a truncation to a few modes or restricted to certain compactifications - my motivations are more primitive • in fact our development of this theory arose from thinking about the issue of locality in theories of quantum gravity more generally, and in particular what sorts of prime principles we should use in building such a theory • now many would say that string theory, as we usually conceive of it, is already a theory of quantum gravity, so why should we think about replacing it by something else? • is string theory a good enough theory of quantum gravity? Metastrings and Locality 2 Generalized Geometry - string theory, at the very least, is rather successful in generating interesting things to think about - at this workshop, there has been a focus on building up (non-) geometric structures that might underly the robust dualities of the theory - a paradigm that we have seen is obtained by considering toroidal compactifications — attempts to think about stringy states (such as winding modes) in a geometric context, in such a way that dualities are realized linearly. • in particular one focuses on some small sub-sector of the theory, at weak coupling, and attempts to describe it in field theoretic terms Metastrings and Locality 3 - although it is interesting to explore the possibilities, forcing string theory to behave like field theory (or quantum to behave as classical) will likely miss the most remarkable features • string theory was developed (in its perturbative definition) as an S- matrix theory for particle states in a flat space-time background CFT consistency Space-time consistency (symmetry, mutual locality, modular inv) (causality, unitarity, locality) • we have a well-defined quantum theory, with an overlay of target space interpretation - I’m going to ask: is there a mistake there? — is that interpretation built in too deeply? • the guiding principle will be locality • we’ll obtain a new string theory in which the target is no longer space- time T duality Born reciprocity − $ Metastrings and Locality 4 Locality - locality plays a central guiding role in modern physics • local effective field theories, locality of interactions, asymptotic states • structure of the renormalization group — separation of scales - QM possesses a variety of non-local features • Heisenberg ∆x∆p 1 ~ ≥ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference - we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes Metastrings and Locality 5 Locality - locality plays a central guiding role in modern physics • local effective field theories, locality of interactions, asymptotic states • structure of the renormalization group — separation of scales - QM possesses a variety of non-local features • Heisenberg ∆x∆p 1 ~ ≥ 2 • entanglement, Bell inequalities, measurement respects local causality • Aharonov-Bohm, interference } - we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes Metastrings and Locality 5 Locality - locality plays a central guiding role in modern physics • local effective field theories, locality of interactions, asymptotic states • structure of the renormalization group — separation of scales - QM possesses a variety of non-local features • Heisenberg ∆x∆p 1 ~ ≥ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference - we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes Metastrings and Locality 5 Locality - locality plays a central guiding role in modern physics • local effective field theories, locality of interactions, asymptotic states • structure of the renormalization group — separation of scales - QM possesses a variety of non-local features • Heisenberg ∆x∆p 1 ~ ≥ 2 • entanglement, Bell inequalities, measurement • Aharonov-Bohm, interference - we expect that any theory of quantum gravity will involve non-locality • in GR, no local diff-invariant observables • gravity is holographic • strings (or D-branes, …) are non-local probes messing with locality must be done very carefully! Metastrings and Locality 5 - there is ample evidence in string theory that the local space-time picture must break down • “string bits” (Klebanov-Susskind): ∆X2 λ2 h µi⇠ • regulate string on lattice: find that rms distance diverges logarithmically • strings become long and space-time wants to discretize • at odds with picture of small strings in smooth space-time • hard scattering (Gross-Mende): • finite temperature (Atick-Witten): • string dualities Metastrings and Locality 6 Locality: absolute vs. relative Absolute locality: space-time is independent of its probes • this is the version of locality that we use implicitly • appropriate to classical, point-like probes Relative locality: space-time depends on the (quantum) nature of probe • e.g., energy or other quantum numbers Metastrings and Locality 7 Locality: absolute vs. relative Absolute locality: space-time is independent of its probes • this is the version of locality that we use implicitly • appropriate to classical, point-like probes Relative locality: space-time depends on the (quantum) nature of probe • e.g., energy or other quantum numbers high energy probe low energy probe STM visualization of quantum crystals at different (S. Davis) probe energies disordered ordered translation inv’t Metastrings and Locality 7 - same system, but physics looks different when probed at different energies! • the classical question is it ordered or disordered? is ill-defined in quantum theory! - if this analogy (crystal space-time, electrons probes) makes any sense, suggests that! if locality is relative, the proper! setting for quantum geometry is a phase space P • need both space-time and energy-momentum for complete description • we should expect energy-momentum to be curved • this is unusual — usually we think of this as a linear space - suggests that there ought to be two scales a • will induce in string theory, and interpret A X /λ X = P λ Ya/" 2 " = ~, = ↵0 ✓ ◆ " Metastrings and Locality 8 Born Reciprocity - in perturbative string theory, the elementary probes are strings, and the degrees of freedom are maps X :⌃ M ! - on the other hand, it is a familiar feature of quantum mechanics that a choice of basis for Hilbert space is immaterial • e.g., for particle states, q is just as good as p | i | i • one choice may be preferred given a choice of observables • e.g., for atomic systems: • interaction with light use energy basis ! • for material properties use position basis ! • a change of basis is accomplished by Fourier transform - applying this to string theory, we interpret the identification of M as space-time as breaking Born reciprocity [M. Born, 1930’s] • by taking the target to be space-time, we’ve chosen a basis Metastrings and Locality 9 Born Reciprocity - this choice of basis is directly associated with asking the string theory to give local effective field theories - but the underlying structure of the string is BR-symmetric • diff constraints: 1 1 H : p2 + δ2 = N + N˜ 2 2 2 − D : p δ = N N˜ · − - what is non-generic is solving the constraints in a particular way • this is a choice of boundary condition — periodicity of the string • this is where locality is built in • of course, in toroidal compactifications, we see the full structure and the resulting duality Metastrings and Locality 10 Born Reciprocity and T-duality - usually, in toroidal compactifications, we interpret short distance (radius) as long distance in a dual space-time - however we can directly relate this to a Fourier transform - consider a string state ↵0p = dX 2 C ⇤ iSP [X ]/λ Z x(σ) [x(σ)] = [DX Dg ] e X =x δ = dX Z |@⌃ C - define a Fourier transform of this state by Z i/ xµdy ˜ [y(σ)] [Dx(σ)]e ~ @⌃ µ [x(σ)] ⌘ Z R - extending y ( σ ) to the world sheet, we integrate out X to obtain a dual Polyakov path integral 2 iS [Y ]/" δ = ↵0 dY ˜ [y(σ)] = [DY Dg ] e− P ⇤ ZC Y @⌃=y Z | p = dY ZC Metastrings and Locality 11 Born Reciprocity and T-duality - usually, in toroidal compactifications, we interpret short distance (radius) as long distance in a dual space-time - however we can directly relate this to a Fourier transform - consider a string state ↵0p = dX 2 C ⇤ iSP [X ]/λ Z x(σ) [x(σ)] = [DX Dg ] e X =x δ = dX Z |@⌃ C - define a Fourier transform of this state by Z i/ xµdy ˜ [y(σ)] [Dx(σ)]e ~ @⌃ µ [x(σ)] ⌘ Z R - extending y ( σ ) to the world sheet, we integrate out X to obtain a dual Polyakov path integral 2 iS [Y ]/" δ = ↵0 dY ˜ [y(σ)] = [DY Dg ] e− P ⇤ ZC Y @⌃=y Z | p = dY δ and p exchange their roles ZC Metastrings and Locality 11 Born Reciprocity and T-duality λ - the Fourier transform induces
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