Lightcone String from String Bits Charles Thornthorn/munich17.pdfmigrate from one part of a chain to...

Lightcone String from String Bits

Charles Thorn

University of Florida

Munich, Lightcone2017

String on the Lightcone

xµ(σ, τ), µ = 0, 1, . . . , d+ 1 embeds worldsheet in spacetime.Pµ conjugate to xµ, and Pµ =

∫

dσPµ is Minkowski momentum.

Phase space constraints from Nambu action (T0×Area):

x′ · P = 0, P2 + T 20 x

′2 = 0, ′ =∂

∂σ

Lightcone Coordinates

x± = (x0 ± x1)/√2, x = {xk}, k = 2, · · · , d+ 1

and similarly for Pµ.

Lightcone parametrization: x+′ = 0 and P+ = 1, so

P− =1

2(P2 + T 2

0x′2), x−′ = x

′ ·P

0-1

Hamiltonian

Because P+ = 1, P+ gives range of σ.

H = P− =

∫ P+

0

dσ1

2(P2 + T 2

0x′2),

∫ P+

0

dσx′ ·P = 0

Dynamics that of “ordinary” Newtonian string in d dimensions,with speed of sound T0.

EG =1

2

∑

n

2|n|πT0P+

= (∞)P+ − πT06P+

Mild form of Holography: almost all of x−,P+ redundant

0-2

Two Aspects of Lorentz Invariance

1. When d ≥ 2, [M−j ,M−k] = 0 requires d = 24 (D = 26)

2. For all d, 3 Ground string vertex ∝ (P+1 P

+2 P

+3 )−d/48, but

Lorentz invariance requires power be −1/2, i.e. d=24.

+2

+1

P + P++

1 2

P

P

0-3

Lightcone parameters have cast string dynamics in a remarkableway. A single string moving in d+1 spatial dimensions is formu-lated as a Newtonian string moving in d dimensions. The lostdimension has been reduced to P+ as a measure of the quan-tity of this Newtonian string. Amazingly, when d = 24 thisunassuming nonrelativistic d dimensional system behaves as aLorentz invariant relativistic quantum string moving in 25 spa-tial dimensions.

For me, this situation points to a fascinating alternative to quan-tum field theory to describe relativistic quantum physics. In par-ticular it may shed light on the mysteries of quantum gravity.

As a first step, we hypothesize that the string is not truly con-tinuous but comprised of discrete entities, “String Bits”. Andthe quantity of string is measured in bit number P+ = mM ,M = 0, 1, 2, . . . This renders x− completely redundant!

0-4

Emergent transverse space

Spin waves on long chains have the character of continuous co-ordinates. In this way transverse space can arise for long chainsof string bits, though each bit exists in only a finite number ofstates.

Spaceless string bit models can provide underlying microscopicdynamics of superstring and hence also can shed lighton quantum gravity.

String bit models provide a concrete way to thinkabout “emergent space”, an idea inspired by black holefirewalls and other difficulties in quantum gravity.

Since space is absent in the underlying dynamics, locality mustbe a derived property of the emergent space.

0-5

Bosonic String to Bits(1977, with Giles)[9]

H =1

2

∫ P+

0

dσ[

p2 + T 2

0x′2]

H =1

2m

M∑

k=1

[

p2n + T 2

0 (xn+1 − xn)2]

0-6

Ex =T0m

M−1∑

n=1

sinnπ

M=T0m

cotπ

2M∼ 2T0M

mπ− πT0

6Mm

0-7

Eθ = −T0m

M−1∑

n=1

sinnπ

M= −T0

mcot

π

2M∼ −2T0M

mπ+

πT06Mm

0-8

Holography(1991)[4]

String bits do not sense x−. But for a chain of many bits

EG − αM ∼ β

Mm+O(M−3) ∼ β

2P+

So long chains do sense x−, which is conjugate to P+.

M bit chain with M → ∞ behaves as though it could movein a longitudinal direction.

0-9

d = D − 2 transverse dimensionality; s θ dimensionality

• d > s: chains unstable

• d < s: long chains stable and string will emerge

• d = s: marginal stability and spacetime supersymmetryfor d = s = 8.

Study of d < s of obvious interest, even though full Lorentzinvariance is absent.

0-10

String Bits (1991)[4, 16]

Through string joining and splitting a string bit on one chain canmigrate from one part of a chain to another part or to anotherstring: Near neighbor interactions only an approximation forfree strings. Therefore string bits should be described in secondquantization language. Each bit is an adjoint in an internal U(N)color space. String perturbation theory is 1/N expansion.

[aβα, aδγ ] = δδαδ

βγ , α, β, γ, δ = 1 · · ·N

H = Traa+g

NTraaaa

0-11

Action of H on singlet states

Singlet states spanned by the (nonorthogonal) set:

Tral1Tral2 · · ·Traln |0〉,∑

i

li =M

Linearly independent for fixed M , sufficiently large N ! Then

HTraM |0〉 =MTraM |0〉+gMTraM |0〉+ g

NM

∑

l

TralTraM−l|0〉

Defines spectrum of H on singlets at continuous 1/N

0-12

Superstring Bits

Fermionic bits needed for stability of system.

Introduce superstring bit annihilation operators

(φ[a1···an])βα , n = 0, . . . , s

n even : Boson, n odd : Fermion

ai: s-valued spinor index

No transverse space; No longitudinal space

Finite number = 2sN2 degrees of freedom. A single superstringbit: quantum system with finite # of states

0-13

Chains of String Bits at N = ∞Consider a Hamiltonian with structure

H =1

N

∑

V ABCDTrφAφBφCφD

[(φA)βα, (φB)

δγ ] = δδαδ

βγ δAB, α, β, γ, δ = 1 · · ·N

Action of H on single trace states:

[

1

NV ABCDTrφAφBφCφD

]

TrφF1· · · φFM

|0〉

=M∑

k=1

V ABFk+1FkTrφF1· · · φFk−1

φAφB · · · φFM|0〉+O(N−1)

Note how N → ∞ singles out n.n. interactions on a linear chain

0-14

Illustrative Hamiltonian for s = 1.

This toy model should give insight into string formation andholography

φβα, (φ1)βα → aβα, b

βα

There is no space: this is ordinary QM with 2N2 d.o.f.

Superstring bit Hamiltonian becomes:

H =2

NTr

[

(a2 − ib2)a2 − (b2 − ia2)b2 + (ab+ ba)ba+ (ab− ba)ab]

H commutes with the Grassmann odd operator

Q = Tr(abeiπ/4 + bae−iπ/4)

This supersymmetry guarantees that states of opposite statisticshave the same energy, enabling statistics waves with excitationenergies of order O(1/M) for large bit number.

0-15

Color singlet Energy spectrum for N = ∞

Introduce superfield: Φ(θ) =∑

n,akφ[a1···an]θ

a1 · · · θan

H

∫

dsMθTrΦ(θ1) · · ·Φ(θM )|0〉Ψ(θ1, . . . , θM )

=

∫

dMθTrΦ(θ1) · · ·Φ(θM )|0〉hΨ(θ1, . . . , θM ) +O(1/N)

h on next page

0-16

h =M∑

k=1

s∑

a=1

[

− 2iθakθak+1 − 2i

d

dθak

d

dθak+1

− 2θakd

dθak+1

−2θak+1

d

dθak− 2 + 4θak

d

dθak

]

Cyclic constraint: Ψ(θ1, . . . , θM ) = (−)s(M−1)Ψ(θ2, . . . , θM , θ1)

Ground Energy:

EG = −sT0m

M−1∑

n=1

sinnπ

M= −sT0

mcot

π

2M∼ −2T0Ms

mπ+πT0s

6Mm(1)

Comment:Excitations of order O(1/M) are color singlets only.Nonsinglet energy levels lie above EG, separated by O(1) gap.

0-17

Message so far:

Chain with lowest energy per bit has M → ∞: And becausethere are excitation energies of order O(1/M) the chain behavesas continuous string!

Moreover, a single string is stable.

The string excitation spectrum is that of s pairs of Grassmannodd worldsheet fields θaL(σ, τ), θ

aR(σ, τ).

For s = 8, this is the Grassmann odd sector of the Green-Schwarzlightcone superstring.

0-18

Where can we find d transverse coordinates?

Possibility 1: s→ s+ 2d and bosonize each pair of extra θ’s:Compactified boson coordinate, nonzero R

Possibility 2. Add another index to φ: Heisenberg anisotropicchain.

0-19

Bosonizing θ’s

Puzzle: zero point energy of bosonic worldsheet fields is supposedto be −πdT0/(6Mm), not +2πdT0/(6Mm).

Resolution of puzzle: Bosonization coordinate is compactifiedon a finite radius circle, with either KK momenta or windingnumbers odd. Thus KK momentum terms and winding numberterms can’t both vanish.

These facts are captured by Jacobi formula:

q1/62∞∏

n=1

(1 + q2n)2 = q−1/12∞∑

m=−∞

q(m+1/2)2∞∏

n=1

1

1− q2n

which equates the partition function of two integer-moded fermionson left with 1 integer moded boson on the right.

0-20

We must do something more sophisticated to get large compact-ification radius and the possibility of 0 winding number.

Possibility 2

Anisotropic Heisenberg spin chain, −1 < ∆ < 1.

H =C

2

M∑

k=1

(

σ1kσ

1k+1 + σ2

kσ2k+1 +∆σ3

kσ3k+1

)

(2)

C can be +ve or −ve. For M even these are equivalent, but notfor M odd. String bit models admit all M .

Energy eigenvalues solved by Bethe ansatz, and explicitly forlarge M (Yang and Yang, 1966).

0-21

Energy spectrum for Heisenberg spin chain

δEf ∼ |C|2π sinµµ

[

−π6+π − µ

4Q2 +

P 2

π − µ+ 2π(NL +NR)

]

1

M

δEaf ∼ C2π sinµ

π − µ

[

−π6+µ

4Q2 +

1

µP 2 + 2π(NR +NL)

]

1

M,

∆ = − cosµ, δE = E − EB

EB : “bulk” part of ground energy exactly proportional to M .

Distinct cases: Let k, l be integers

f : C < 0, Q = k, P = lπ

af : C > 0, Q = k, P =

{

lπ Q even

(l + 1/2)π Q odd

0-22

Compare to energy compactified string coordinate: x ≡ x+ L:

P− =1

2

∫ P+

0

dσ[

P2 + T 20 x

′2]

given by

P− =T0P+

[

−π6+

2π2

4L2T0(2k)2 +

L2T02

l2 + 2π(NL +NR)

]

Exact match of Q even sector, with

L2 =2π

T0

π

π − µor

2π

T0

π

µ(3)

By varying µ, can reach L in range 2π√α′ < L <∞

Comment: For C > 0, decompactification L → ∞ sends odd Qsector to infinite energy.

0-23

Transverse coordinates emerge as flavor waves on string bit chain.

We have seen that the continuum limit of a critical Heisenbergspin chain describes a spatial coordinate compactified on a circleof radius R = 2π/

√

2T0(π − µ) where ∆ = − cosµ [19].

One could append a two valued “flavor index” to the super-string bit annihilation operator for each transverse dimension:φ[a1···am] → φf1···fd[a1···am], with fi = 1, 2.

Then it is possible to design the string bit Hamiltonian to pro-duce the Heisenberg Hamiltonian on long chains in the limitN → ∞.

0-24

What happens when N is finite? Can we still take M → ∞?

Songge Sun and Gaoli Chen have studied the N dependence ofenergy eigenstates with fixed bit number M = 3, 4, 5, . . . up toM = 11.

Comments:

• Multi-trace states not linearly independent for fixed N =integer..

• Energies can become complex when N is not an integer.

• For finite integer N there is probably an upper limit toM < 2N + 1 for a stable ground state.

• If so N must be enormous in order that Lorentz violationsdue to finite M do not contradict experiment.

• Correspondingly string interactions must be incrediblyweak.

0-25

Finite Temperature and Hagedorn Phase

A string bit system at low energy has a spectrum which lookslike that of a system of strings moving in space at low energy.But at high energy the system looks very different, with spacemelted away,

Temperature is a popular tool to probe these different behav-iors. Let’s examine the partition function Z = Tre−βH for thesimplest string bit model, with s = 1.

0-27

At low temperature and large N , Z is well approximated by thepartition function for an ideal gas of strings.

lnZ =∑

f

ln(1 + e−βǫf )−∑

b

ln(1− e−βǫb)

=∞∑

n=1,odd

1

n

∑

b

e−nβǫb + (−)n−1∑

f

e−nβǫf

≡∞∑

n=1,odd

1

nz(nβ)

The even terms cancel by supersymmetry.

The string partition functions z(β) diverges when β < βH =T−1H =

√

π/(6T0), with TH the Hagedorn temperature. But thiscan’t happen in the string bit model. Therefore the string pic-ture must break down at a temperature below TH , and replacedby a picture more accurately described as string bits. SouravRaha, a graduate student at UF, is making good progress inunderstanding how this crossover in behaviors occurs.

0-28

1/N Corrections and String Interactions

Recap Kinematics of Superstring Bits

• Superstring bit annihilation operators

(φ[a1···an])βα ; n = 0, . . . , s; ai = 1, . . . , s; α, β = 1, . . . , N

• n even : Boson, n odd : Fermion

• No transverse space; No longitudinal space; Only time

• Finite number = 2sN2 degrees of freedom.

• Hamiltonian for general s in arXiv:1407.8144.

0-29

Hamiltonian for s = 1

φ, φ1 → a, b

Superstring bit Hamiltonian::

H =2

NTr

[

(a2 − ib2)a2 − (b2 − ia2)b2 + (ab+ ba)ba+ (ab− ba)ab]

This is ordinary QM with 2N2 d.o.f.

Supersymmetry:

Q = Tr(abeiπ/4 + bae−iπ/4), [Q,H] = 0, Q2 = Tr[

aa+ bb]

≡M

0-30

H splits or fuses traces

Superfield: ψ(θ) = a+ θb.

T (θ1, · · · θM ) ≡ Trψ(θ1) · · ·ψ(θM )

Fission:

HT (θ1θ2 · · · θM )|0〉 = hT (θ1θ2 · · · θM )|0〉

+1

N

∑

l 6=k,k+1

hklT (θl · · · θk)T (θk+1 · · · θl−1)|0〉

Fusion:

HT (θ1 · · · θK)T (φ1 · · ·φL)|0〉 = (hθ + hφ)T (θ)T (φ)|0〉+1

N(Fission Terms)

+1

N

K∑

k=1

L∑

l=1

[

hklT (θk+1 · · · θkφl · · ·φl−1) + hlkT (θk · · · θk−1φl+1 · · ·φl)]

|0〉

0-31

First Quantized Hamiltonian

hkl = −2iθkθl − 2id

dθk

d

dθl+ 2θk

d

dθl+ 2θl

d

dθk+ 2− 4θk

d

dθk

On integration by parts we shall use

hkl = −2iθkθl − 2id

dθk

d

dθl− 2θk

d

dθl− 2θl

d

dθk− 2 + 4θk

d

dθk

Then h =∑

k hk,k+1 and h =∑

k hk,k+1.

0-32

Perturbation in 1/N

At zeroth order we first solve the eigenvalue problem

hψr(θ1, · · · , θMr) = Erψr(θ1, · · · , θMr

)

and then change the single trace operators to energy basis

Tr =

∫

dθ1 · · · dθMrT (θ1, · · · , θMr

)ψr(θ1, · · · , θMr)

ψr(θ1, . . . , θMr) = (−)Mr−1ψr(θ2, . . . , θMr

, θ1)

Define the conjugate to ψr, denoted ψr such that∫

dθ1 · · · dθMrψs(θ1, · · · , θMr

)ψr(θ1, · · · , θMr) = δrs

and so the completeness relation in each M sector is written∑

r

ψr(θ1, · · · , θMr)ψr(φ1, · · · , φMr

) = δ(θ − φ)

0-33

In energy basis the action of H on single traces state becomes

HTr|0〉 = ErTr|0〉+1

N

∑

s,t

TsTt|0〉Vstr

Vstr =∑

l 6=k,k+1

∫

dθψs(θl · · · θk)ψt(θk+1 · · · θl−1)hklψr(θ1, · · · , θM )

From cyclic symmetry

ψs(θl · · · θk)hkl = ψs

(

: hkl : +EG

Ms

)

∼ ψs (: hkl : +α)

where EG ∼ αMs +O(1/Ms).

: hkl : in Vstr is negligible in continuum limit:

Vstr ∼ α∑

l 6=k,k+1

∫

dθψs(θl · · · θk)ψt(θk+1 · · · θl−1)ψr(θ1, · · · , θMr)

0-34

Hamiltonian can also fuse pair of traces into one trace:

(HTsTt|0〉)Fuse ≡ Tr|0〉Wrst

where

Wrst =∑

k,l

∫

dθdφ

[

ψr(θk+1 · · · θkφl · · ·φl−1)hkl

+ψr(φl+1 · · ·φlθk · · · θk−1)hlk

]

ψs(θ1 · · · θMs)ψt(φ1 · · ·φMt

)

Again, in the continuum limit hkl and hlk can be replaced by α,

Wrst ∼ 2α∑

k,l

∫

dθdφψr(θk+1 · · · θkφl · · ·φl−1)ψs(θ1 · · · θMs)ψt(φ1 · · ·φMt

)

The double sums in V andW simply include all ways of splittinga chain in two or of joining two chains into one.

0-35

Continuum Limit

∑

k,l

→ 1

m2

∫

dσdσ′

Additional 1m from P− = T0

4mH.

Smooth continuum limit of interactions implies overlap ∼M−3,which fixes critical dimension of the emergent string, whether ornot there is Lorentz invariance.

0-36

Lowest order energy shifts.

|E〉 =∑

r

Tr|0〉C1r +

∑

st

TsTt|0〉C2st +

∑

stu

TsTtTu|0〉C3stu + · · ·

require that (H − E)|E〉 = 0:E.g. suppose only C1

r with common Er are 6= 0 at N = ∞:

C2st =

1

E − Es − Et

1

N

∑

r

VstrC1r +O(N−2)

(E − Er)C1r =

1

N2

∑

st

Wrst1

E − Es − Et

∑

u

VstuC1u +O(N−3)

Then

∆ru =1

N2

∑

st

Wrst1

Er − Es − EtVstu

Determines lowest order energy shifts to the level Er (G. Chen).

0-37

Eigenstates of h

Clifford algebra:

Sk = θk +d

dθk, Sk = i

(

θk − d

dθk

)

h =∑

k

[

−iSkSk+1 + iSkSk+1 − iSkSk+1 + iSkSk+1 + 2iSkSk

]

{Sk, Sl} = 2δkl, {Sk, Sl} = 2δkl, {Sk, Sl} = 0

To find the eigenvalues of h, Fourier transform

Bn =1√M

M∑

k=1

Ske−2πikn/M , Bn =

1√M

M∑

k=1

Ske−2πikn/M

{Bm, Bn} = 2δm+n,M , {Bm, Bn} = 2δm+n,M , {Bm, Bn} = 0

0-38

Define

∆ = 8 sinπn

Mn = 0, 1, . . . ,M − 1

energy lowering and raising operators

Fn = Bn cosπn

2M+ iBn sin

πn

2M, Fn = F †

M−n

then

[h, Fn] = −∆Fn, [h, Fn] = ∆Fn, {Fn, Fm} = 2δn+m,M

Ground Energy: Fm|G〉 = 0, m > 0

P−G = −T0

m

M−1∑

n=1

sinnπ

M∼ −2T0M

mπ+

πT06Mm

0-39

The Overlap

Divide theM spin variables into L (k = 1, . . . L) and K =M−L(k = L + 1, . . . ,M) variables. Then for each subset we definemodes

Sk =1√L

L−1∑

n=0

B(1)n e2πikn/L, 1 ≤ k ≤ L

Sk =1√K

M−L−1∑

n=0

B(2)n e2πi(k−L)n/K , L+ 1 ≤ k ≤M

and likewise for Sk, with a tilde on corresponding B’s.

For zero modes we take B0, B0 and define

b0 = B(1)0

√

K

M−B

(2)0

√

L

M, b0 = B

(1)0

√

K

M− B

(2)0

√

L

M

0-40

Relating Fm to F(1)n , F

(2)n

Let fk be the M − 1 operators

f0 ≡ 1

2(b0 + ib0),

F(1)n√2,

F(2)n√2, {fk, f†l } = δkl

Then

Fm =√2M−2∑

n=0

(fnCmn + f†nSmn)

where, e.g.

Cmn1 = − 1√ML

1− e−2πimL/M

1− e−2iπ(n/L−m/M)cos

(nπ

2L− mπ

2M

)

Smn2 =1√MK

1− e−2πimL/M

1− e2iπ(n/K+m/M)cos

( nπ

2K+mπ

2M

)

0-41

Ground state of single chain on two chain space

|G〉 = exp

1

2

∑

k,l

Mklf†kf

†l

|0〉[det(I +MM †)]−1/4

where fk|0〉 = 0 for k = 0, . . . ,M−2. with Fm|G〉 = 0 equivalentto

CmnMnl + Sml = 0.

From CM = −S we compute

det(I +MM †) =det(CC† + SS†)

det(CC†)

0-42

MATLAB Numerics

det(CC† + SS†) = 1

detCC† ∼ 0.9290

[KLM ]1/6

(

L

M

)α(K,L)/3−2/3(K

M

)α(L,K)]/3−2/3

α(K,L) ≡ M

K− L

M, M = K + L

Compare to bosonic string overlap:

detXX† =2.1528

[KLM ]1/6

(

L

M

)−α(K,L)/3(K

M

)−α(L,K)/3

This result, apart from the numerical factor, can be understoodbased on the conformal mapping properties of the worldsheet.

0-43

Clarification

For superstring (d = s) overlap involves:

detCC† detXX† =2.0000

[KLM ]1/3

(

KL

M2

)−2/3

= 2.0000M

KL

Green-Schwarz: “bosonic and spinor determinants cancel”

Actually means: dependence of an offshell vertex on the interac-tion time e−ia∆P−

has no measure contribution because

(∆P−)zero−point ∝s− d

6∆

1

2P+→ 0

for d→ s, as Poincare supersymmetry requires.

But the cancellation is actually incomplete, and left-over part isessential to account for the eventual Poincare invariance.

0-44

Assembling String Models

Vertex Insertions from String Bits

∆X = Xk+1 −Xk ∼ m∂X∂σ means ∆X ∼M−1 nominally.

Sk,Γk ∼ √m(S(σ),Γ(σ)) means Sk,Γk ∼M−1/2 nominally

Fission/Fusion singularity modifies these expectations:

Insertion Enhancement Net

∆X M1/2 M−1/2

S M1/2 M0

Γ M1/4 M−1/4

0-45

Critical dimension

Everything done for s = 1 extends to general s w/o difficulty!

Overlap Scaling Insertion Scaling

VX M−d/8 ∆X M−1/2

VS M−s/8 S M0

VΓ M−d/16 Γ M−1/4

Smooth continuum limit: vertex ∼M−3 for K/M,L/M fixed.

Type Total Vertex Critical DimensionBosonic String VX d = 24IIB Superstring ∆Xi∆XjPij(S)VXVS d = s = 8RNS String (Γ ·∆X)2VΓVX d = 8Protostring VS s = 24

0-46

New Proto-String

• d = 0, s = 24 gives finite continuum limit w/o insertions.

• String in 1 space and 1 time dimension (x+, x−)

• This is a pure Grassmann analog of the bosonic string.

• Stable with no massless particles

• Degree of freedom count matches the superstring:

Bosonize 16 Grassmann fields =⇒ 8 (compactified) scalar fields.

• Superstring a deformation of this proto-string?

• A solid foundation for string?

0-47

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−10

−5

0

5

10

15

1/N

E

Energy,M=3

0-48

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−2

0

2

4

6

8

10

12

1/N

Norms of Energy Eigenstates,M=3

Nor

m

0-49

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−10

0

10

20

30

40Energy, M=5

1/N

E

0-50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

0

2

4

6

8

10Norms of Energy Eigenstates, M=5

1/N

Nor

m

0-51

References

[1] S. Sun and C. B. Thorn, Phys. Rev. D 89 (2014) 10, 105002[arXiv:1402.7362 [hep-th]]. [2]

[2] C. B. Thorn, JHEP 1411 (2014) 110 [arXiv:1407.8144 [hep-th]].

[3] C. B. Thorn, Phys. Rev. D 92 (2015) 6, 066007[arXiv:1507.03036 [hep-th]].

[4] C. B. Thorn, In *Moscow 1991, Proceedings, Sakharovmemorial lectures in physics, vol. 1* 447-453, and [arXiv:hep-th/9405069].

[5] G. ’t Hooft, Found. Phys. 43 (2013) 46.

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[6] P. Goddard, C. Rebbi, C. B. Thorn, Nuovo Cim. A12

(1972) 425-441. P. Goddard, J. Goldstone, C. Rebbi andC. B. Thorn, Nucl. Phys. B 56 (1973) 109.

[7] G. ’t Hooft, Nucl. Phys. B342 (1990) 471; “On the Quan-tization of Space and Time,” Proc. of the 4th Seminar on

Quantum Gravity, 25–29 May 1987, Moscow, USSR, ed. M.A. Markov, (World Scientific Press, 1988); “Dimensional re-duction in quantum gravity,” gr-qc/9310026.

[8] S. Mandelstam, Nucl. Phys. B 64 (1973) 205. Nucl. Phys. B69 (1974) 77.

[9] R. Giles and C. B. Thorn, Phys. Rev. D 16 (1977) 366.

[10] L. Brink and H. B. Nielsen, Phys. Lett. B 45 (1973) 332.

[11] F. Gliozzi, J. Scherk and D. I. Olive, Nucl. Phys. B122 (1977) 253; P. Ramond, Phys. Rev. D 3 (1971) 2415;

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A. Neveu and J. H. Schwarz, Nucl. Phys. B 31 (1971) 86;A. Neveu, J. H. Schwarz and C. B. Thorn, Phys. Lett. B35 (1971) 529. C. B. Thorn, Phys. Rev. D 4 (1971) 1112;A. Neveu and J. H. Schwarz, Phys. Rev. D 4 (1971) 1109.

[12] K. Bardakci and M. B. Halpern, Phys. Rev. D 3 (1971)2493.

[13] M. B. Green and J. H. Schwarz, Nucl. Phys. B 181 (1981)502.

[14] O. Bergman and C. B. Thorn, Phys. Rev. D 52 (1995) 5980[hep-th/9506125].

[15] G. ’t Hooft, Nucl. Phys. B72 (1974) 461.

[16] C. B. Thorn, Phys. Rev. D 20 (1979) 1435.

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[17] C. B. Thorn, “Substructure of string,” talk to Strings 96:Current Trends in String Theory, 15-20 Jul 1996. Santa Bar-bara, California [arXiv: hep-th/9607204].

[18] O. Bergman and C. B. Thorn, Phys. Rev. D 52 (1995)5997 [hep-th/9507007]. Phys. Rev. Lett. 76 (1996) 2214 [hep-th/9512107]. Nucl. Phys. B 502 (1997) 309 [hep-th/9702068].C. B. Thorn, Phys. Rev. D 56 (1997) 6619 [hep-th/9707048].

[19] R. Giles, L. D. McLerran and C. B. Thorn, Phys. Rev. D17, 2058 (1978).

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