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Sourav Raha

[WWS21] Hypergraphs as chains of string bits

Sourav Raha

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Gauge multiplets to lightcone strings
Sourav Raha
University of Florida

Abstract: In this project, we use hypergraphs to represent the color lines of a quantum system that has adjoint matter. We derive the replacement rules corresponding to the action of a quartic Hamiltonian. We discuss the supersymmetric extension of this formalism and how it is related to lightcone quantized strings.

Introduction

The simplest non-trivial operation in quantum mechanics is the commutator:







It is the cornerstone of Heisenberg’s uncertainty principle. It also comes in handy for constructing the eigenvalues of the quantum harmonic oscillator where the ladder operators satisfy

[a,

]1

In the operator formalism, the matrix elements of any quantum operator can be written as

<0|…aa…

…aa…

…|0>

If the ground state is normalized, computing such elements is equivalent to repeated applications of the commutator relation of the ladder operators. This can push all the annihilation operators to the right of all the creation operators. This is referred to as normal ordering in literature.

Normal ordering

We might use string substitution systems to implement normal ordering. For example, we show how a an expression like

can be normal ordered

[◼]	MultiwaySystem

[{"aA""","aA""Aa"(*,"bB""","bB""Bb","aB""Ba","bA""Ab"*)}({SelectFirst[#,First[#]==("aA""")&],SelectFirst[#,First[#]==("aA""Aa")&]}&),"aaAAaaAA",90,"StatesGraph"(*,"EvolutionGraph"*)(*,"IncludeStatePathWeights"True,VertexLabels"VertexWeight"*)]

Out[]=

One can see that the normal ordering produces 4 distinct terms in the end: including an empty (vacuum) state. One question could be: how to obtain operator subsequences that simplify to the vacuum? The following multiway system shows how this can be achieved:

In[]:=

[◼]	MultiwaySystem

[{"""aA","Aa""aA"(*,"bB""","bB""Bb","aB""Ba","bA""Ab"*)},"",3,"StatesGraph"(*,"IncludeStatePathWeights"True,VertexLabels"VertexWeight"*)]

Out[]=

The leftmost sequences would correspond to the norms of the oscillator eigenstates.

StringJoin@@@Tuples[{"A","B"},2]ToLowerCase/@%Outer[StringJoin,%,%%]

Out[]=

{AA,AB,BA,BB}

Out[]=

{aa,ab,ba,bb}

Out[]=

{{aaAA,aaAB,aaBA,aaBB},{abAA,abAB,abBA,abBB},{baAA,baAB,baBA,baBB},{bbAA,bbAB,bbBA,bbBB}}

Adjoint multiplets

However, one could ask: how to deal with situations where the particle is a multiplet of some internal degree of freedom? String substitution systems do not have the power to capture such extra degrees of freedom. For example, we might have adjoint multiplets





where

μ,ν

are ‘color’ indices and take values from

1…N

. In that case, we leverage the power of hypergraphs. The hypergraphs represent the state of a system, with each expression representing a color line and the atoms representing the operators that share a particular color index (i.e. the color index is contracted between the two operators). Here we represent an initial state that represents a single trace state of

particles

|0>

In[]:=

<<SetReplace`

ToExpression["A"<>ToString[#]]&/@Range[10];ToExpression["c"<>ToString[#]]&/@Range[10];init=Join[{RotateRight[%%]},{%},{%%}]//TransposeHypergraphPlot[init,VertexLabelsAutomatic]

Out[]=

{{a10,c1,a1},{a1,c2,a2},{a2,c3,a3},{a3,c4,a4},{a4,c5,a5},{a5,c6,a6},{a6,c7,a7},{a7,c8,a8},{a8,c9,a9},{a9,c10,a10}}

Out[]=

The simplest non-trivial Hamiltonian is the one with a quartic interaction:

This term annihilates two quanta followed by creation of two quanta. The application of this Hamiltonian does not change the total number of particles. All it does is to identify certain color indices. Its action has been implemented using the following set of rules:

In[]:=

evo=ResourceFunction["WolframModel"][<|"PatternRules"{{{c1__,a1_},{a1_,c2__},{c3__,a2_},{a2_,c4__}}Module[{w,b1,no,b2,ea,s},{{c1,a1,w,b1},{c3,a2,s,a1,c2},{b1,no,b2},{b2,ea,a2,c4}}](*,{{c1__,a1_},{a1_,c2__},{c3__,a2_},{a2_,c4__}}Module[{s,b1,b2,ea,w,no},{{c1,a1,s,a2,c4},{b2,ea,a1,c2},{c3,a2,w,b1},{b1,no,b2}}]*)}|>,init,50]evo["ExpressionsEventsGraph"]evo["EventsStatesPlotsList"]

Out[]=

WolframModelEvolutionObject

Generations: 5

Events: 7



Out[]=

One could also take the inner product with the bra vector

<0|tr

|ψ>

to compute the vacuum-to-vacuum amplitudes. These amplitudes are in powers of

. Each expression in the final state represents an index cycle of the color degree of freedom and contributes a factor of

. The quartic vertex contributes a factor of

-1

. Thus one may look at any final state and obtain the amplitude of the corresponding physical process. The

dependence of these vacuum amplitudes corresponds to the topological expansion of strings.

In[]:=

inner=ResourceFunction["WolframModel"][<|"PatternRules"{{{c1__,a1_},{a2_,c4__}}Module[{s},{{c1,a1,s,a2,c4}}](*,{{c1__,a1_},{a1_,c2__},{c3__,a2_},{a2_,c4__}}Module[{s,b1,b2,ea,w,no},{{c1,a1,s,a2,c4},{b2,ea,a1,c2},{c3,a2,w,b1},{b1,no,b2}}]*)}|>,evo[1],1]inner["ExpressionsEventsGraph"]inner["EventsStatesPlotsList"]

Out[]=

WolframModelEvolutionObject

Generations: 1

Events: 5



Out[]=

In[]:=

inner["FinalStatePlot"]inner["FinalState"]//Length

Out[]=

Connection to string bits

String bits have a super - creation operator given by

(η)

where

μ,ν

are ‘color’ indices and take values from

1…N

is a Grassmann number, i.e.

{η,η}0

. The state space of

string bits is then spanned by

∏

(

)|0>

with

denoting the site number on a one - dimensional lattice. One can show that the large

limit of action the Hamiltonian:

H

2mN

tr[(

-i

)aa-(

-i

)bb+(

)ba+(

)ab]

on long (i.e.

M∞

), closed chains (i.e. single trace states) of string bits reproduces the light cone

corresponding to a subcritical string. This string has only one Grassmann world sheet field (no transverse coordinates) and one longitudinal coordinate,



-i

∫

dσ [S(σ)

′

(σ)-



(σ)

′



(σ)]

In the continuum limit chains the following identifications are made:

kσ

m

and



S(σ)

.The

are (Majorana-Weyl) spinors and are related to the Grassmann coordinates as





i

and

{

}

k,l

Future prospects

Extend the amplitude calculation developed here to include fermionic excitations. This would require dealing with Grassmann numbers and their anti-commuting properties.

Look to establish the correspondence with light cone

by analysing the spectrum of

for finite bit number

(M)

and finite number of colors

(N)

Complete project work


Keywords

◼

String bits

Acknowledgment

Mentor: Tobias Canavesi

References

◼

Bergman, O., & Thorn, C. B. (January 01, 1995). String bit models for superstring. Physical Review. D, Particles and Fields, 52, 10, 5980.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.52.5980

POSTED BY: Sourav Raha

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[WWS21] Hypergraphs as chains of string bits

Introduction

Normal ordering

Adjoint multiplets

Connection to string bits

Future prospects

Complete project work

Keywords

Acknowledgment

References

Complete project work
