Desperate spacetimes call for desperate measures
Introduction
Desperate spacetimes call for desperate measuresAn argument for superspace realism
Tushar Menon
Balliol College, Oxford
University of Geneva, 30th November 2016
Desperate spacetimes call for desperate measures
Introduction
Introduction
Why care about spacetime?
Why care about supersymmetry?
Why care about superspace?
Three idealisations/simplifications–(i) Rigid SUSY (ii)Non-extended (N = 1) SUSY (iii) Unbroken SUSY
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Contents
1 Introduction
2 Supersymmetry and superspaceSupersymmetrySuperspace
3 What is spacetime?Mathematical spacetimesPhysics and spacetimeStructuring spacetime
4 Why is superspace spatiotemporal?
5 Conclusion
6 AppendicesDirect and semidirect productsGratuitous mathematics
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Supersymmetry
What is Supersymmetry?
Supersymmetry is a transformation between bosons andfermions that is structure preserving.
Can be thought of dynamically—a set of transformationswhich leaves invariant the form of the equations of motion.(As a consequence, they map solutions to solutions)
Or a spacetime symmetry—a set of transformations whichleaves invariant certain geometric objects characteristic ofspacetime structure.
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Superspace
What is superspace?
In ordinary spacetime, events are coordinatised usingquadruples of real numbers. (More generally, n-tuples)
A field is a map from this space of events (independentvariables) to some mathematical space.
A bosonic field is a map from this space to a space ofcommuting operators, while a fermionic field is a map to aspace of anticommuting operators.
In the classical limit, we can consider analogues of these twotypes of fields—classical bosonic fields are maps into realnumbered tuples, classical fermionic fields are maps intoanticommuting-numbered tuples.
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Superspace
What is superspace?
Imagine combining the two spaces in which our fields taketheir values into a single space—one with a commuting halfand an anticommuting half. Elements of this space are knownas supernumbers. The commuting supernumbers are denotedas Rc and the anticommuting ones as Ra.
The real numbers themselves are elements of the commutingsubalgebra of supernumbers. So the spacetime manifold,diffeomorphic to R4, is a special kind of generalised manifold,which we refer to as superspace.
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Superspace
What is superspace?
A field theory containing interacting bosonic and fermionicfields can be seen as a map from ordinary spacetime to asupernumber space.
We can generalise this further, and consider maps fromsuperspace to a supernumber space. These are known assuperfields.
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Superspace
What is superspace?
Instead of considering an interacting boson/fermion fieldtheory as maps from ordinary spacetime to a supervectorspace, let us think of it as a superfield, which takes values inRc .
Since a superfield’s domain is superspace, we incorporate theanticommuting nature of the fermionic fields directly into ourbase space.
More specifically, if our superfield formally looks like this:Φ : R4|4 → Rc , then it can be expanded in a coordinate basiswhich includes anticommuting supernumber-valuedcoordinates, θ and θ
In this coordinate basis, a generic superfield looks like this:Φ(x , θ, θ) = φ(x) + ψα(x)θa + ψβ(x)θβ + Dαβ(x)θαθβ
Desperate spacetimes call for desperate measures
Supersymmetry and superspace
Superspace
The question
So far, we have a way of showing that some of the degrees offreedom of a field can be brought down to the base manifold.
Can we give some argument for why these degrees of freedomspatiotemporal? Especially given that the SUSY generatorsanticommute as:
{Qa,Q†b} = (σµ)abPµ
What is it to be spatiotemporal?
Desperate spacetimes call for desperate measures
What is spacetime?
Table of Contents
1 Introduction
2 Supersymmetry and superspaceSupersymmetrySuperspace
3 What is spacetime?Mathematical spacetimesPhysics and spacetimeStructuring spacetime
4 Why is superspace spatiotemporal?
5 Conclusion
6 AppendicesDirect and semidirect productsGratuitous mathematics
Desperate spacetimes call for desperate measures
What is spacetime?
Mathematical spacetimes
Mathematical and physical spacetimes
We need to distinguish between two senses in which we talkabout spacetimes
We first need the appropriate (minimally structured)mathematical object to be able to express our dynamics.Thisgives us mathematical spacetimes.
We then need to identify features of the physical world thatneed to be appropriately represented. This gives us physicalspacetimes.
Desperate spacetimes call for desperate measures
What is spacetime?
Mathematical spacetimes
Mathematical spacetimes and fibre bundles
The geometric structure needed to represent generalrelativistic spacetime must have at least enough structure todefine metric compatibility and torsion-freedom.
To define torsion freedom, we need to define torsion—and forthat, we need to be able to symmetrise the connection.
Desperate spacetimes call for desperate measures
What is spacetime?
Mathematical spacetimes
Finding spacetime
The notion of spacetime only needs to make sense at the alevel where we need to preserve empirical coherence.
Spatiotemporal geometry need not be fundamental.
Look at the dynamics. It’s all in the dynamics.
Space and time are intuitively background structures.Background independence just localises the notion of abackground structure.
Desperate spacetimes call for desperate measures
What is spacetime?
Physics and spacetime
How to write a field theory with a dynamical symmetry
Decide on the domain for your field theory—this is usuallytaken to be Minkowski spacetime (i.e. a manifolddiffeomorphic to R4), but we cannot assume that right now
Introduce a (dynamical) symmetry which you would like yourtheory to exhibit.
Use the elements of this symmetry group to construct theassociated Lie algebra (perhaps hoping that the Lie Group issimply connected)
Construct representations of the Lie algebra
Use the matrices so obtained to study the objects whichtransform under the symmetry
Construct invariant Lagrangians using covariant combinationsof these objects and use them to derive equations of motion
Desperate spacetimes call for desperate measures
What is spacetime?
Physics and spacetime
Transformation properties
Construct representations of the Lie algebra
Use the matrices so obtained to study the objects whichtransform under the symmetry
Not all objects transform the same way under symmetrygroups.
Looking at representations of the Poincare group, one seesthat fields transform in different representations of the Lorentzsubgroup—this is how we introduce spinor, scalar and vectorfields into our theory
But all fields transform in the same representation of thetranslation group. And this group has four generators.
Desperate spacetimes call for desperate measures
What is spacetime?
Physics and spacetime
Transformations of fields
The symmetry group of free fields is always G × ISO(1, 3)
The symmetry group of the Standard Model isSU(3)× SU(2)× U(1)× ISO(1, 3)
Fields may transform under different groups, or under differentrepresentations of the same group
For example, a lepton field transforms trivially under SU(3)
But there is no field that transforms in a differentrepresentation of the translation group than all the others.
So let us use the translation group to identify the independentvariables that we treat as spacetime variables
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Knox on inertial frames
We can now talk about how to structure the space ofindependent variables
Look at the geometric characterisation of spacetime structurefor clues—let us start with inertial structure.
Inertial frames are frames with respect to which force freebodies move with constant velocities.
The laws of physics take the same form (a particularly simpleone) in all inertial frames.
All bodies and physical laws pick out the same equivalenceclass of inertial frames (universality).
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Inertial frames
The last two of Knox’s points are guaranteed by the StrongEquivalence Principle (SEP), but Read and Brown’s argumentsuggests that the SEP is violated, even by minimally coupledfields.
This doesn’t mean we can’t have nice things (i.e. inertialstructure).
Inertial trajectories are geodesics.
These trajectories can be defined metrically or with referenceto an affine connection. (And assuming metric compatibility,will coincide with the affine geodesics).
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Inertial frames (contd.)
All the relevant information about the metric (certainlyenough to derive its geodesics) is contained in the group of itsisometries.
So all we need for local inertial structure is local Poincareinvariance.
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Too much structure?
What are the intuitive characteristics of spacetime thatinertial frame functionalism captures?
Causal structure, Forces, Change, Motion and Dynamics
Even if this is too rich a notion in general, it is not too rich toclassify superspace as spatiotemporal
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Semidirect product structure
If all there is to being a spacetime degree of freedom is that itis characterised by the universal transformation behaviour ofthe fields, why is, for example, SO(987, 497) not a spacetimesymmetry group?
This points to a further subtlety, a further piece of structure,required of the algebraic characterisation. The concept of asemi-direct product provides the requisite structure.
Consider the symmetry group of the Standard Model again.Notice how three of the four groups are simple.
Desperate spacetimes call for desperate measures
What is spacetime?
Structuring spacetime
Summary
We identified, as the appropriate mathematical structure torepresent our spacetime a set with enough structure as toallow us to define torsion.
We then used arguments from the transformation propertiesof fields to identify the defining characteristics ofspacetime—the independent variables ‘associated with’ (in asense made precise) translations.
Finally, we structured the spacetime by identifying one of thenormal subgroups of the total symmetry group of the theoryas the spacetime symmetry group—the group thatdecomposed into a semidirect product.
We’ve reversed the order of priority of fields and spacetime, atleast methodologically. Further metaphysical claims couldpotentially be made to follow.
Desperate spacetimes call for desperate measures
Why is superspace spatiotemporal?
Table of Contents
1 Introduction
2 Supersymmetry and superspaceSupersymmetrySuperspace
3 What is spacetime?Mathematical spacetimesPhysics and spacetimeStructuring spacetime
4 Why is superspace spatiotemporal?
5 Conclusion
6 AppendicesDirect and semidirect productsGratuitous mathematics
Desperate spacetimes call for desperate measures
Why is superspace spatiotemporal?
Generating translations
We are used to thinking of both boosts and translations beinggenerated by differential operators of some description,through a Taylor series. For translations, for example,exponentiating the generator of translations is equivalent toperforming a Taylor expansion as follows:
φ(xµ)→ eaµ∂µφ(x) = φ(xµ) +
1
2aµ∂µφ(x) + ... = φ(xµ + aµ).
Desperate spacetimes call for desperate measures
Why is superspace spatiotemporal?
Generating supertranslations
In the superspace case, exponentiating a supercharge allowsone to reexpress the superfield as having been translated byan infinitesimal amount along theanticommuting-supernumber-parametrised dimensions as well:
Φ(x , θ, θ)→ Φ
(x + a +
i
2ζσµθ − i
2θσµζ, θ + ζ, θ + ζ
).
We arrive at this by treating the θ terms as spacetimecoordinates, and shifting the superfield in the same way wewould an ordinary field:
Φ(x , θ, θ) = U(x , θ, θ)Φ(0)U†(x , θ, θ)
With the following requirement on generators:
U(x , θ, θ) = exp(ix · P) · exp(iθ · Q) · exp(i θ · Q)
Desperate spacetimes call for desperate measures
Why is superspace spatiotemporal?
Structuring superspace
The symmetry group of a supersymmetric theory is the directproduct of some symmetry group with the Super-Poincaregroup.
The Super-Poincare group splits into the semidirect productof the Lorentz group and the supertranslation group.
The symmetry group has the right structure to play the roleof a spacetime symmetry group. The Poincare group gives usthe structure associated with spacetime—causality, forces,change and dynamics. So does the Super-Poincare group.
Desperate spacetimes call for desperate measures
Conclusion
Table of Contents
1 Introduction
2 Supersymmetry and superspaceSupersymmetrySuperspace
3 What is spacetime?Mathematical spacetimesPhysics and spacetimeStructuring spacetime
4 Why is superspace spatiotemporal?
5 Conclusion
6 AppendicesDirect and semidirect productsGratuitous mathematics
Desperate spacetimes call for desperate measures
Conclusion
Dynamical Relativity and Spacetime Functionalism
On the functionalist approach, spacetime refers to the degreesof freedom of a theory which suffice in specifying local inertialstructure.
On the algebraic approach, we recover Knox’s geometricfunctionalism, but we are also at liberty to make claims aboutthe spatiotemporal nature of degrees of freedom that are notmanifestly spatiotemporal. For example, superspace.
Desperate spacetimes call for desperate measures
Conclusion
Conclusions
We need not worry about identifying spatiotemporal degreesof freedom in our fundamental ontology (it’s not even clearthat this is possible unproblematically).
Insofar as we would like to recover our phenomenologicalexperience of spacetime, we can look to the dynamics withsome prescription.
The prescription that I suggest involves two constraints—(i)universality of behaviour of fields. (ii) semi-direct structure onthe symmetry group.
It allows us to make sense of a spatiotemporal interpretationof superspace.
More broadly, this allows me to complement the sorts ofprojects that the emergent spacetime and quantum gravitypeople are involved in.
Desperate spacetimes call for desperate measures
Conclusion
Thank you.
Desperate spacetimes call for desperate measures
Appendices
Table of Contents
1 Introduction
2 Supersymmetry and superspaceSupersymmetrySuperspace
3 What is spacetime?Mathematical spacetimesPhysics and spacetimeStructuring spacetime
4 Why is superspace spatiotemporal?
5 Conclusion
6 AppendicesDirect and semidirect productsGratuitous mathematics
Desperate spacetimes call for desperate measures
Appendices
Direct and semidirect products
Appendix 1: Direct and semidirect products
Consider two groups, N and H. Their semidirect product isconstructed by first taking the Cartesian product of theunderlying sets. The binary operation, denoted as • is definedusing a homomorphism ϕ from H to Aut(N) (in other words,we find a representation of H on N), denoted as nϕ. Aproduct of two elements of G can now be written as
(n1, h1) • (n2, h2) = (n1ϕ(h1)(n2), h1h2) = (n1ϕh1(n2), h1h2).
Desperate spacetimes call for desperate measures
Appendices
Gratuitous mathematics
Appendix 2: Gratuitous mathematics
δ(Boson) ∝ (Fermion) (1)
δ(Fermion) ∝ (Boson) (2)
δφ = ζ · χ, (3)
δχ = Cζφ, (4)
δχ = −i(∂µφ)σµiσ2ζ∗ (5)
δβδαφ = δβ(ζ · χ), (6)
δβδαφ = C ′ζ∂µφ. (7)