Lectures on Twistors

0.48 MB·English

by  Itzhak Bars

1
USC-06/HEP-B1 hep-th/0601091
Lectures on Twistors
1
ItzhakBars
DepartmentofPhysicsandAstronomy
6 UniversityofSouthernCalifornia,LosAngeles,CA90089-0484,USA
0
0 Abstract
2
IntheselecturesIwilldiscussthefollowingtopics
n
a
J Twistorsin4flatdimensions.
•
4
– Masslessparticles,constrainedphasespace(xµ,pµ)versustwistors.
1
– Physicalstatesintwistorspace.

1
v Introductionto2T-physicsandderivationof1T-physicsholographsandtwistors.
1 •
9
– Emergentspacetimes&dynamics,holography,duality.
0
1 – Sp(2,R)gaugesymmetry,constraints,solutionsand(d,2).
0
– Globalsymmetry,quantizationandtheSO(d,2)singleton.
6
0 – Twistorsforparticledynamicsind dimensions,particleswithmass, rela-
/ tivistic,non-relativistic,incurvedspaces,withinteractions.
h
t
- Supersymmetric2T-physics,gaugesymmetries&twistorgauge.
p
•
e
h – CouplingX,P,g,gaugesymmetries,globalsymmetries.
:
v – Covariantquantization,constrainedgenerators&representationsofGsuper.
i
X – Twistor gauge: supertwistors dual to super phase space. Examples in
d=4,6,10,11.
r
a
Supertwistorsandsomefieldtheoryspectraind=4,6.
•
– SuperYang-Millsd=4,N=4;Supergravityd=4,N=8.
– Self-dualtensorsupermultipletandconformaltheoryind=6.
Twistorsuperstrings
•
– d+2viewoftwistorsuperstringind=4.
– Worldsheetanomaliesandquantizationoftwistorsuperstring.
– Openproblems.
1Lectures delivered at the “2005 Summer School on String/M Theroy” in Shanghai, China, and the
InternationalSymposiumQTS4,“QuantumTheoryandSymmetriesIV”,Varna,Bulgaria.
2 1 TWISTORSIND=4FLATDIMENSIONS
1 Twistors in d=4 flat dimensions
Amasslessspinlessrelativisticparticlein4space-timedimensionsisdescribedbythe
action
1
S(x,p)= dτ ∂ xµp ep p ηµν . (1)
τ µ µ ν
− 2
Z (cid:18) (cid:19)
It hasa gaugesymmetryunderthe transformationsδ e = ∂ ε(τ),δ xµ = ε(τ)pµ,
ε τ ε
δ p = 0. The generator of the gauge symmetry is p2/2,and it vanishes as a conse-
ε µ
quenceoftheequationofmotionforthegaugefieldδS/δe=p2/2=0.Thisequation
ofmotionisinterpretedasdemandingthatthesolutionspacemustbegaugeinvariant
(sincethegeneratormustvanish).
Inthecovariantquantizationofthissystemonedefinesthephysicalstatesasthose
thatsatisfytheconstraintp2 φ = 0,sothattheyaregaugeinvariant. Acompleteset
| i
of physical states is found in momentum space k on which the gauge generator is
| i
simultaneously diagonal with the momentum operator p k = k k , and p2 k =
µ µ
| i | i | i
k k2 = 0. The probability amplitude of a physical state in position space xφ =
| i h | i
φ(x) satisfies the condition xp2 φ = 0 which gives the Klein-Gordon equation
h | | i
∂2φ(x) = 0. The generalsolution is a superpositionof plane waves, which are the
probabilityamplitudesofphysicalstateswithdefinitemomentum
d4k
Generalsolution:φ(x)= (2π)4δ k2 a(k)eik·x+h.c. (2)
Z
(cid:0) (cid:1)(cid:2) (cid:3)
Planewavewithdefinitemomentumkµ:φ (x)= xk eikx, k2 =0. (3)
k ·
h | i∼
A similar treatment for spinning particles leads to the spinning free field equations,
suchastheDiracequation,Maxwellequation,linearizedEinsteinequation,etc.
1.1 Twistors
Thefollowingshowsseveralwaysofsolvingtheconstraintp2 =0ork2 =0thatenter
intheseequations
p2 =0: p0 =± p~2 or pαβ˙ =±(λλ†)αβ˙ = √12pµ(σµ)αβ˙ (4)
p− =p2p/2p+ 2 2Hermitian,rank1,uptophaseλ eiφλ
⊥ × →
In thesecondform, the matrixp is constructedfromtwo complexnumbersλ ,λ
αβ˙ 1 2
thatformadoubletofSL(2,C)=SO(3,1)
λ λ λ λ λ 1 p0+p3 p ip
p=±(cid:18)λ12(cid:19)(λ∗1 λ∗2)=±(cid:18)λ21λ∗1∗1 λ21λ∗2∗2(cid:19)= √2(cid:18)p1+ip2 p10−−p32(cid:19) (5)
Thishasautomaticallyzerodeterminantdet(p)=(λ λ )(λ λ ) (λ λ )(λ λ )=
1 ∗1 2 ∗2 − 1 ∗2 2 ∗1
0= p0+p3 p0 p3 (p ip )(p +ip )=p2 p~2,whichimposesthedesired
− − 1− 2 1 2 0−
solutionpµp = 0automatically. Notethattheoverallphaseeiφ ofλ dropsout,so
µ α
(cid:0) (cid:1)(cid:0) (cid:1)
thematrixp reallyhasonly3realparameters,asitshould.
αβ˙
1.1 Twistors 3
ThereaderisremindedofabitofgrouptheoryforSL(2,C)=SO(3,1)
λ 1,0 εαβ orεα˙β˙ = 0 1
spinors: α 2 , invarianttensors: 10
(cid:26) λ¯α˙ ≡(cid:0) λ†α˙(cid:1) 0,21 ( metric,raise/low(cid:16)er−indi(cid:17)ces
(σµ)αβ˙ =(cid:0)(1,~σ(cid:1))αβ˙; (σ¯µ)α˙β =(−1,~σ)α˙β; (12,12)= ((210,,01))×((01,,120)
vectors:  pµ : pαβ˙ = √12pµ(σµ)αβ˙, p¯α˙β = √12pµ(σ¯µ)α˙β, 2 × 2
 xµ : xαβ˙ = 1 xµ(σ )αβ˙, x¯α˙β = 1 xµ(σ¯ )α˙β
√2 µ √2 µ
Penrose[2][3]suggestedasecondspinorµα˙ andintroducedthe“incidencerelation”
whichdefinesxasbeingroughlythe“slope”ofa“line”inspinorspace
µα˙ = ix¯α˙βλ , a“line”inspinorspace. (6)
β
−
Finally a twistor is defined as Z = µα˙ , A = 1,2,3,4,that bundles togetherµ
A λα
andλasaquartet. IfµsatisfiesthePen(cid:16)rose(cid:17)relation,thenthepairµ,λisequivalentto
thethephasespaceofthemasslessparticle
µα˙ ( ix¯λ)α˙ on-shell
Z = = − (7)
A (cid:18)λα(cid:19) λα !⇔ psphaacsee (xµ,pµ)
Althoughnotmanifest,themasslessparticleactionabovehasahiddenconformal
symmetry SO(4,2). This symmetry can be made manifest through the twistor since
SO(4,2) = SU(2,2) and the quartet Z can be classified as the fundamentalrepre-
A
sentation4ofSU(2,2).Thisnon-compactgrouphasametricwhichcanbetakenas
C = 01 =σ 1.Usingthemetricwedefinetheotherfundamentalrepresentation
¯4ofSU1(02,2)an1d×relateittothecomplexconjugateofZ asfollows
A
(cid:0) (cid:1)
Z¯A =Z†C = λ†α˙ µ†α = λ†α˙ iλ†x¯ α , C =σ1×1 (8)
So Z¯AZ is invariant unde(cid:16)r SU(2,2(cid:17)). W(cid:16)e rem(cid:0)ind t(cid:1)he(cid:17)reader that the 4 and 4¯ of
A
SU(2,2)correspondtothetwoWeylspinorsofSO(4,2).Now,withµasgivenabove,
wehave
Z¯AZA =λ†α˙µα˙ +µ†αλα =−iλ†x¯λ+iλ†x¯λ=0. (9)
So,byconstructiontheZ are4constrainedcomplexnumbers.Butwecanreverse
A
thisreasoning,andrealizethatthe definitionoftwistorsis justthestatementthatZ
A
isaquartetthathasanoverallirrelevantphaseandthatisconstrainedbyZ¯AZ = 0.
A
Then the form of µ in terms of λ can be understood as one of the possible ways of
parameterizingasolution. Thesolutionµα˙ = ix¯α˙βλ isinterpretedasthemassless
β
−
particle.Thisistheconventionalinterpretationoftwistors.
However, recently it has been realized that there are many other ways of param-
eterizing solutions for the same Z in terms of phase spaces that have many other
A
differentinterpretations[8]. Foranysolution,ifwecountthenumberofindependent
realdegreesoffreedom,wefind
1realconstraint sameas
Independent: (8realZ) =6real= (10)
− 1realphase 3~x+3~p
(cid:18) (cid:19)
4 1 TWISTORSIND=4FLATDIMENSIONS
This is the right numbernot only for the massless particle, but also the massive par-
ticle, relativistic or non-relativistic, in flat space or curved space, interacting or non-
interacting.
Next,wecomputethecanonicalstructureforthepair Z ,Z¯A ,andwefindthat
A
itisequivalenttothecanonicalstructureinphasespaceforthemasslessparticle,iffwe
(cid:0) (cid:1)
usethesolutionµα˙ = ix¯α˙βλ
β
−
L=iZ¯A∂ Z =iλ¯ ∂ µα˙ +iµ¯α∂ λ
τ A α˙ τ τ α
=λα†˙∂τ(x¯λ)α˙ − λ†x¯ α∂τλα
=λα†˙ ∂τx¯α˙β λβ(cid:0)=T(cid:1)r(p∂τx¯)=pµ∂τxµ
Sothecanonicalpairs Z ,(cid:0)iZ¯A or(cid:1) λ ,iµ α or(xµ,p )areequivalentaslongas
A α † µ
theysatisfytherespectiveconstraintsZ¯AZ = 0andp2 = 0. Ifweusesomeofthe
A
(cid:0) (cid:1) (cid:0) (cid:1)
othersolutionsgivenin[8]thenthecorrectcanonicalstructureemergesforthemassive
particle,etc.,allfromthesametwistor(seebelow).
Just like the constraint p2 = 0 followed from an action principle in Eq.(1), the
constraints Z¯AZ = 0 can also be obtained from the following action principle by
A
minimizingwithrespecttoV
S(Z)= dτ Z¯AiD Z 2hV = dτ Z¯Ai∂ Z +VZ¯AZ 2hV .
τ A τ A A
− −
Z (cid:0) (cid:1) Z (cid:0) (cid:1)(11)
Here D Z = ∂ Z iVZ is a covariant derivative for a U(1) gauge symmetry
τ A τ A A
−
Z (τ) Z (τ) = eiω(τ)Z (τ).Thegaugesymmetryispreciselywhatisneeded
A → A′ A
toremovetheunphysicaloverallphasenotedabove.
For spinning particles, an extra term 2hV is included in the action (missing in
−
formerliterature).ThistermisgaugeinvariantbyitselfundertheU(1)gaugetransfor-
mationofV.Wehavebeendiscussingthespinlessparticleh = 0,buttwistorscanbe
generalizedtospinningparticlesbytakingh=0.Theequationofmotionwithrespect
toV givestheconstraintZ¯AZ = 2h.Ifthe6 twistortransformformasslessparticles,
A
appropriatelymodifiedtoincludespin,isusedtosolvethisconstraint[2][3][4],then
hisinterpretedasthehelicityofthespinningmasslessparticle. Butifthemoregen-
eraltransformsin [8]are used, then h is nothelicity, butis aneigenvalueofCasimir
operatorsofSU(2,2)inarepresentationforspinningparticles2.
We have argued that the twistor action S(Z) is equivalent to the spinless mass-
less particle action S(x,p) (at least in one of the possible ways of parameterizing
its solutions). But note that S(Z)is manifestlyinvariantunderthe globalsymmetry
SU(2,2). This is the hidden conformal symmetry SO(4,2) of the massless particle
action S(x,p). ApplyingNoether’stheoremwe derivethe conservedcurrent, which
2Thispointwillbediscussedindetailinafuturepaper.
1.2 Physicalstatesintwistorspace 5
inturniswrittenintermsofxµ,p asfollows
µ
1 ix¯λ
JAB =ZAZ¯B− 4ZCZ¯CδAB = −λ λ† iλ†x¯ (12)
(cid:18) (cid:19)
(cid:0) (cid:1)
ix¯λλ x¯λλ x¯ ix¯p x¯px¯ 1
= − † † = = ΓMNL (13)
MN
λαλ†β˙ iλαλ†x¯! (cid:18) p −ipx¯(cid:19) 4i
1 1
= Γ+′ ′L+′ ′ + L Γµν Γ+′L ′µ Γ ′L+′µ (14)
2i − − − 2 µν − µ − − −µ
(cid:18) (cid:19)
Inthelastlinethetraceless4 4matrix ix¯p x¯px¯ isexpandedintermsofthefol-
× p ipx¯
−
lowingcompletesetofSO(4,2)gammam(cid:16)atricesΓM(cid:17)N (M = ,µ,seefootnote(6))
±
10 σ¯µν 0 σ¯µν σ¯[µσν]
Γ+′−′ = − , Γµν = , ≡ (15)
0 1 0 σµν σµν σ[µσ¯ν]
(cid:18) (cid:19) (cid:18) (cid:19) ≡
0σ¯µ 0 0
Γ+′µ =i√2 , Γ ′µ = i√2 , (16)
−
0 0 − σµ0
(cid:18) (cid:19) (cid:18) (cid:19)
ThisidentifiesthegeneratorsoftheconformalgroupLMN asthecoefficients
x2
L+′ ′ =x p, Lµν =xµpν xνpµ, L+′µ =pµ, L ′µ = pµ xµx p. (17)
− −
· − 2 − ·
It can be checked that this form of LMN are the generators of the hidden SO(4,2)
conformal symmetry of the massless particle action. The SO(4,2) transforma-
tions are given by the Poisson brackets δxµ = 1ω LMN,xµ and δpµ =
2 MN
1ω LMN,pµ , and these LMN are the conserved charges given by Noether’s
2 MN (cid:8) (cid:9)
theorem. Furthermore they obey the SO(4,2) Lie algebra under the Poisson brack-
(cid:8) (cid:9)
ets. ThisresultisnotsurprisingoncewehaveexplainedthatS(Z)= S(x,p)viathe
twistortransform.
ThesameSU(2,2)symmetryofthetwistoractionS(Z)hasotherinterpretations
as the hidden symmetry of an assortment of other particle actions when other forms
oftwistortransformisused,asexplainedin[8]. Thisrecentbroaderresultmayseem
surprisingbecauseitiscommonlyunfamiliar.
1.2 Physicalstates intwistorspace
Incovariantquantizationaphysicalstateforaparticleofanyhelicityshouldsatisfythe
helicityconstraint 1(Z Z¯A+Z¯AZ )ψ =2hψ .Thisisinterpretedasmeaningthat
2 A A | i | i
thephysicalstate ψ isinvariantundertheU(1)gaugetransformationgeneratedbythe
| i
constraintthatfollowedfromthetwistoractionS(Z). TheprobabilityamplitudeinZ
space is ψ(Z) Z ψ , so we can write Z¯Aψ(Z) = Z Z¯A ψ = ∂ ψ(Z).
≡ h | i h | | i −∂ZA
Thenthehelicityconstraint 1 Z (Z Z¯A+Z¯AZ )ψ =2h Z ψ producesthephys-
2h | A A | i h | i
icalstatecondition,
∂
Z ψ(Z)=( 2h 2)ψ(Z) (18)
A
∂Z − −
A
6 1 TWISTORSIND=4FLATDIMENSIONS
for a particle of helicity h. So a physicalwavefunctionin twistor space ψ(λ,µ) that
describesaparticlewithhelicityhmustbehomogeneousofdegree( 2h 2)under
− −
the rescalingZ tZ or(µ,λ) (tµ,tλ) [2][3]. Thisis the onlyrequirementfor
→ →
a physical state ψ(Z) in twistor space, and it is easily satisfied by an infinite set of
functions.
Ifweusethetwistortransformformasslessparticlesµ= ix¯λandp = λλ¯,then
−
anyhomogeneousphysicalstateintwistorspaceshouldbeasuperpositionofmassless
particle wavefunctions since p2 = 0 is automatically satisfied. A similar statement
would hold for any of the other twistor transforms given in [8], so a physical state
in twistor space can also be expandedin terms of the wavefunctions2 of the particle
systemsdiscussedin[8][9].
Letusnow considerthe expansionof a physicalstate ψ in termsofmomentum
| i
eigenstates pµ k = kµ k , for a massless particle with k2 = 0. We parameterize
| i | i
k = π π¯ asin Eq.(4), whereπ canbe redefinedupto a phase π eiγπ with-
αβ˙ α β˙ α →
out changing the physical state k . In position space such a physical state gave the
| i
plane wave as in Eq.(3), which we can rewrite as φ (x) = xk exp(ik x) =
k
h | i ∼ ·
exp(iTrx¯ππ¯). The twistor space analogis φ (λ,µ) = Z k = λ,µπ,π¯ . Since
k
h | i h | i
k isacompletesetofstates,itispossibletowriteageneralphysicalstateintwistor
| i
spaceasaninfinitesuperpositionofthe Z k witharbitrarycoefficients,inthesame
h | i
wayasthegeneralsolutionoftheKlein-GordonequationinEq.(2)
ψ(Z)= d2πd2π¯[a(π,π¯) Z π,π¯ +h.c.] (19)
h | i
Z
To determine Z k = λ,µπ,π¯ , first note that the eigenstate of λ is propor-
α
h | i h | i
tional to π , so there must be an overall delta function λ,µπ,π¯ δ( λπ ). The
α
h | i ∼ h i
argument of the delta function is the SL(2,C) invariant dot product defined by the
symbol λπ λ π εαβ.Thevanishingof λπ =0requiresλ π ,henceinthe
α β α α
h i ≡ h i ∝
wavefunction λ,µπ,π¯ wecanreplaceλ = λπ uptoanoverallconstantcsym-
h | i α π α
bolizedbyc= λ.Thisistheratioofeithercomponent λ λ1 = λ2.Sowecanwrite
π π ≡ π1 π2
Z k = λ,µπ,π¯ =δ( λπ )f π,π¯,λ,µ .Nextexaminethematrixelementsof
hthe|twiistohrtran|sformi p h λiλ¯ =0anπdapplytheoperatorsoneithertheketorthe
αβ˙ − α β˙ (cid:0) (cid:1)
braasfollows(λ¯ actsasaderivative ∂ ontheeigenvalueofµβ˙)
β˙ −∂µβ˙
∂
0= Z p λ λ¯ k = k +λ λ,µπ,π¯ (20)
h | αβ˙ − α β˙ | i αβ˙ α∂µβ˙ h | i
(cid:16) (cid:17) (cid:18) (cid:19)
λ ∂ λ
=δ( λπ )π π¯ + f π,π¯, ,µ . (21)
h i α β˙ π∂µβ˙ π
(cid:18) (cid:19) (cid:18) (cid:19)
The solution is f(π,π¯,λ,µ) = g π,π¯,λ exp ππ¯ µα˙ , for any g π,π¯,λ , so
π −λ α˙ π
Z k =δ( λπ )exp ππ¯ µα˙ g π,π¯,λ .Notethattheexponentialisarewriting
h | i h i −λ α˙ (cid:0) π(cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
oftheplanewaveexp(iTrx¯ππ¯)byusingπ = πλandthensettingµ= ix¯λ.
(cid:0) (cid:1) (cid:0) (cid:1) λ −
Finally we determineg π,π¯,λ for a particlewith any helicity h. Accordingto
π
the previous paragraph, since Z k is a physical wavefunction, it should be homo-
(cid:0) h | i(cid:1)
geneous of degree ( 2h 2) under a rescaling (µ,λ) (tµ,tλ). It should also
− − →
1.2 Physicalstatesintwistorspace 7
be phase invariant under the phase transformations π eiγπ, π¯ e iγπ¯ since
−
→ →
the momentum state k labeled by k = π π¯ is phase invariant. The expo-
| i αβ˙ α β˙
nential exp ππ¯ µα˙ is homogeneous as well as phase invariant, while the delta
−λ α˙
function satisfies δ tλeiγπ = t 1e iγδ( λπ ). These considerations determine
(cid:0) h (cid:1) i − − h i
g π,π¯,πλ = πλ −(cid:0)1−2hφh(π(cid:1),π¯),withφh eiγπ,e−iγπ¯ =e−i2hγφh(π,π¯).
Thespecificφ (π,π¯)foreachhelicityaredeterminedasfollows. φ (π,π¯)must
h h
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
have SL(2,C) spinor indicesfor the representation(j ,j ),since for a spinningpar-
1 2
ticle the complete set of labels includesLorentz indices k,j ,j , in addition to
1 2
| ···i
momentum. The chirality of the SL(2,C) labels must be compatible with the spin
j +j = h. So thisdeterminestheLorentzindicesonthe wavefunctionφ (π,π¯)
1 2 h
| |
as well as the coefficientsa(π,π¯) in Eq.(19). Examplesof the overallwavefunction
Z k isgiveninthetablebelow
h | i
hZ|ki=δ(hλπi)exp −πλπ¯α˙µα˙ πλ −1−2hφh(π,π¯).
particle (j ,j ) φ (π,π¯)
1 2 (cid:0) h (cid:1)(cid:0) (cid:1)
scalar (0,0) φ (π,π¯)=1
0
(0,1) ψh=+1/2(π,π¯)=π¯
quark 2 α˙ α´
(21,0) ψαh=−1/2(π,π¯)=πα
Ah=+1(π,π¯)= wαπ¯β˙
gAauµgepotential 21,12 Ahαα=ββ˙˙−1(π,π¯)= πhαππ¯www¯¯βi˙ (22)
fieldstrength (cid:0)(0,1)(cid:1) Fαh˙β=˙+1(π,π¯)=π¯hα´π¯iβ˙
Fµν (1,0) Fαhβ=−1(π,π¯)=παπβ
gh=+2(π,π¯)= π¯α´π¯β˙wγwδ
gmµeνtric (1,1) gααh˙=ββ˙γγ−˙δδ˙2(π,π¯)= παhπππ¯βww¯w¯iγ˙22w¯δ˙
h i
curvature (0,2) Rαh˙=β˙γ+˙δ˙2(π,π¯)=π¯α´π¯β˙π¯γ˙π¯δ˙
Rµνλσ (2,0) Rαh=βγ−δ2(π,π¯)=παπβπγπδ
The field strength F = ∂ A can be written in terms of the gauge potential in
µν [µ ν]
momentumandspinorspaceforanarbitrarycombinationofbothhelicitiesasfollows
Aαβ˙ =a+A+αβ˙(π,π¯)+a−A−αβ˙(π,π¯) (23)
F =k A k A =ε a+F+ (π,π¯)+ε a F (π,π¯) (24)
αβ˙γδ˙ αβ˙ γδ˙ − γδ˙ αβ˙ αγ β˙δ˙ β˙δ˙ − α−γ
which is consistent with the wavefunctions A (π,π¯),F (π,π¯) given in the table
± ±
above. Notethatallwavefunctionsareautomaticallytransversetok = π π¯ under
αβ˙ α β˙
theLorentzinvariantdotproductusingthemetricinspinorspaceεαβ εα˙β˙.
⊕
In field theory computations that use twistor techniques [17], the twistor space
wavefunctions above are used for the corresponding physical external particles with
definitemomentum,uptooverallnormalizations.
8 2 2T-PHYSICS
2 2T-physics
Asmentionedabove,ithasbeendiscoveredrecentlythattherearemanywaysofsolv-
ingthesameconstraintsonthetwistorZ andderiveotherrelationsbetweenµ,λand
A
phasespace[8].Theseothersolutionsdescribenotonlythemasslessparticle,butalso
massiveparticle,relativisticornon-relativistic,inflatspaceorcurvedspace,interact-
ing or non-interacting, as shown in the examples in Fig.1. These new twistors were
discoveredbyusingtwotimephysics(2T-physics)asatechnique.
2T-physicswasalsousedtoobtainthegeneralizationoftwistorstohigherdimen-
sions,tosupersymmetry,andtoD-branes.IntherestoftheselecturesIwillfirstgivea
briefoutlineofthemainaspectsof2T-physicsandthensummarizethesenewresults.
2T-physics: unified emergent space-times & dynamics,
hidden symmetries, holography and duality in 1T-physics
Emergent
spacetime:
Holography: from
Sp(2,R) gauge
(d,2) to (d-1,1).
choices. Some
spinless All images
combination of
holographically
XM,PMis fixed
represent the
as t,H. spinless
same 2T system
Can fix 3 gauges,
but fix 2 or 3
Hidden
symmetry:
Duality: Sp(2,R)
All images
relates one fixed
have hidden
gauge to another
SO(d,2)
symmetry, for
the example.
8
Unification: 2T-physics unifies diverse forms of 1T-physics into a single theory.
Fig.1-2T-physicsind+2descendstomany1T-physicssystemsin(d 1)+1.
−
2.1 Emergent spacetimes & dynamics, holography,duality.
2T-physicscanbeviewedasaunificationapproachforone-timephysics(1T-physics)
systemsthroughhigherdimensions. ItisdistinctlydifferentthanKaluza-Kleintheory
because there are no Kaluza-Klein towers of states, but instead there is a family of
1T systems with duality type relationships among them. The 2T theory is in d+2
dimensions,buthasenoughgaugesymmetrytocompensatefortheextra1+1dimen-
2.2 Sp(2,R)gaugesymmetry,constraints,solutionsand(d,2) 9
sions,sothatthephysical(gaugeinvariant)degreesoffreedomareequivalenttothose
encounteredin1T-physics.
Oneofthestrikinglysurprisingaspectsof2T-physicsisthatagivend+2dimen-
sional2Ttheorydescends,throughgaugefixing,downtoafamilyofholographic1T
imagesin(d 1)+1dimensions. Fig.1belowillustratesafamilyofholographicim-
−
agesthathavebeenobtainedfromthesimplestmodelof2T-physics[6].Theseinclude
interactingaswellasfreesystemsin1T-physics.
It must be emphasized that as a by product of the 2T-physics approach certain
physicalparameters,suchasmass,parametersofspacetimemetric,andsomecoupling
constants appear as moduli in the holographic image while descending from d + 2
dimensionalphasespaceto(d 1)+1dimensionsortotwistors.
−
EachimagerepresentedbytheovalsaroundthecenterinFig.1fullycapturesthe
gaugeinvariantphysicalcontentofauniqueparent2Ttheorythatsitsatthecenter.But
fromthepointofviewof1T-physicseachimageappearsasadifferent1T-dynamical
system. Themembersofsuchafamilynaturallymustobeyduality-typerelationships
among them and share many common properties. In particular they share the same
overallglobalsymmetryind+2dimensionsthatbecomeshiddenandnon-linearwhen
acting on the fewer (d 1)+ 1 dimensions in 1T-physics. Thus 2T-physics unifies
−
many1Tsystemsintoafamilythatcorrespondstoagiven2T-physicsparentind+2
dimensions.
2.2 Sp(2,R) gaugesymmetry,constraints, solutions and(d,2)
Theessentialingredientin2T-physicsisthebasicgaugesymmetrySp(2,R)actingon
phasespaceXM,P in d+2dimensions. Thetwotimelike directionsis notanin-
M
put,butisoneoftheoutputsoftheSp(2,R)gaugesymmetry. Aconsequenceofthis
gaugesymmetry is that position and momentumbecomeindistinguishableat any in-
stant,sothesymmetryisoffundamentalsignificance.ThetransformationofXM,P
M
isgenerallyanonlinearmapthatcanbeexplicitlygiveninthepresenceofbackground
fields [18], but in the absence of backgroundsthe transformation reduces to a linear
doubletactionofSp(2,R)on XM,PM foreachM [5]. Thephysicalphasespace
is the subspace that is gauge invariant under Sp(2,R). Since Sp(2,R) has 3 gener-
(cid:0) (cid:1)
ators, to reach the physical space we must choose 3 gauges and solve 3 constraints.
So, the gauge invariant subspace of d + 2 dimensional phase space XM,P is a
M
phasespacewithsixfewerdegreesoffreedomin(d 1)spacedimensions xi,p ,
i
−
i=1,2, (d 1).
··· − (cid:0) (cid:1)
InsomecasesitismoreconvenientnottofullyusethethreeSp(2,R)gaugesym-
metry parameters and work with an intermediate space in (d 1) + 1 dimensions
−
(xµ,p ), that includes time. This space can be further reduced to d 1 space di-
µ
−
mensions xi,p byaremainingone-parametergaugesymmetry.
i
Therearemanypossiblewaystoembedthe(d 1)+1or(d 1)phasespacein
(cid:0) (cid:1) − −
d+2 phase space, and this is doneby makingSp(2,R)gaugechoices. In the result-
ing gauge fixed 1T system, time, Hamiltonian, and in general curved spacetime, are
emergent concepts. The Hamiltonian, and therefore the dynamics as tracked by the
emergenttime,maylookquitedifferentinonegaugeversusanothergaugeintermsof
theremaininggaugefixeddegreesoffreedom.Inthisway,aunique2T-physicsaction
10 2 2T-PHYSICS
givesrisetomany1T-physicssystems.
A particle interacting with various backgrounds in (d 1)+1 dimensions (e.g.
−
electromagnetism,gravity,highspinfields,anypotential,etc.),usuallydescribedina
worldlineformalismin1T-physics,canbeequivalentlydescribedin2T-physics.
Thegeneral2Ttheoryforaparticlemovinginanybackgroundfieldhasbeencon-
structed[18].Foraspinlessparticleittakestheform
1
S = dτ X˙iMP AijQ (X,P) , (25)
M ij
− 2
Z (cid:18) (cid:19)
where the symmetric Aij(τ) , i,j = 1,2, is the Sp(2,R) gauge field, and the three
Sp(2,R)generatorsQ (X(τ),P (τ)),whichgenerallydependonbackgroundfields
ij
that are functions of (X(τ),P(τ)), are required to form an Sp(2,R) algebra. The
backgroundfieldsmustsatisfycertainconditionstocomplywiththeSp(2,R)require-
ment. Aninfinitenumberofsolutionstotherequirementcanbeconstructed[18]. So
any 1T particle worldline theory, with any backgrounds, can be obtained as a gauge
fixedversionofsome2Tparticleworldlinetheory.
The 1T systems which appear in the diagram above are obtained by considering
thesimplestversionof2T-physicswithoutanybackgroundfields. The2Tactionfora
“free”2Tparticleis[5]
1 1
S = dτ D XMXNη εij = dτ X˙MPN AijXMXN η .
2T 2 τ i j MN − 2 i j MN
Z Z (cid:18) (cid:19)
(26)
HereXM = XM PM ,i=1,2,isadoubletunderSp(2,R)foreveryM,thestruc-
i
ture D XM = ∂ XM AjXM is the Sp(2,R) gaugecovariantderivative, Sp(2,R)
τ i (cid:0) τ i (cid:1)− i j
indicesareraisedandloweredwiththeantisymmetricSp(2,R)metricεij,andinthe
last expression an irrelevant total derivative (1/2)∂ (X P) is dropped from the
τ
− ·
action. ThisactiondescribesaparticlethatobeystheSp(2,R)gaugesymmetry,soits
momentumandpositionarelocallyindistinguishableduetothegaugesymmetry. The
XM,PM satisfytheSp(2,R)constraints
(cid:0) (cid:1) Q =X X =0: X X =P P =X P =0, (27)
ij i j
· · · ·
thatfollowfromtheequationsofmotionforAij. Thevanishingofthegaugesymme-
try generators Q = 0 implies that the physical phase space is the subspace that is
ij
Sp(2,R)gaugeinvariant. Theseconstraintshavenon-trivialsolutionsonlyifthemet-
ric η has two timelike dimensions. So when position and momentumare locally
MN
indistinguishable,tohaveanon-trivialsystem,twotimelikedimensionsarenecessary
asaconsequenceoftheSp(2,R)gaugesymmetry.
Thusthe XM,PM inEq.(26)areSO(d,2)vectors,labeledbyM = 0,1,µor
′ ′
M = ,µ,andµ=0,1, ,(d 1)orµ= ,1, ,(d 2),withlightconetype
′
± (cid:0) (cid:1) ··· − ± ··· −
definitions of X ′ = 1 X0′ X1′ and X = 1 X0 X3 . The SO(d,2)
± √2 ± ± √2 ±
(cid:16) (cid:17) (cid:0) (cid:1)