Consistent String in any Dimension D > 2 Norbert DragonHannover2. Mai 2013Quantum Theory and UnitarityIn onsistent: Dis rete Momentum EigenstatesFailure of the BRST Constru tionStone and von Neumann ex lude States of the Covariant StringMa key Theorem: Indu ed RepresentationsConsistent String in any Dimension?Who ares?

Relativisti Quantum Theory and UnitarityQuantum theories give probability distributions for results of measurements

w(i; A;) = j h�i i j2h�i �ii h i ; ;�i 2 H ; w(i; A;�j) = Æij :Wigner: Symmetries are realized by unitary or antiunitary transformations.Relativisti is a quantum theory, if a unitary representation (of the onne ted partof the over) of the Poin ar�e group a ts on it's Hilbert spa e.Textbook string theory is no relativisti quantum theory, be ause it la ks a Hilbertspa e, e.g. Ay = A is unde�ned and string al ulations yield no probabilities.Fixing the gauge and solving the onstraints before quantization yields a onsistentquantum string at least in D � 26 dimension.

Dis rete S alar Produ thp p0i ?= Æp;p0) Lorentz transformations a t ompletely dis ontinuous or not at all.ji = j0i 0 + j0i ; j0i =Pi jpii i 6= 0 w.l.g .Hi little group of jpii, dimHi < dim SO(1;D- 1) .Ea h pair jpii ; jpji de�nes a left oset Hij = gijHj of transformations, whi hmap jpji to jpii. Ea h Hij and SijHij have measure 0 in SO(1;D- 1) .The omplement of SijHij has full measure. Therefore, in ea h neighbourhood ofthe unit element, there are elements g whi h map ea h jpii to jgpii 6= jpji 8j .) h0 10i 6= 0; h0 g0i = 0 dis ontinuous, no in�nitesimal generators.All group averages vanish, 8� : R� dg h0 g0i = R� dg = 0 :

The BRST Constru tionQuantum theories with spin � 1 have Fo k spa esh� i = Z dD-1pgmn�m �(p)n(p)

with inde�nite (but nondegenerate) g = diag(1; 1; : : : 1| {z }p ;-1;-1 � � �- 1| {z }q ) .signature: sigg = p- q ; siggMN = siggM siggN

If a BRST operator Q, Q2 = 0, ommutes with P, then it's ohomology inheritsa metri g ohQ with the same signature,sig(g oh(Q)) = sig(g) :

A BRST onstru tion Q is su essful, if the ohomology of Q de�nes a Hilbertspa e, j sig(g oh(Q))j = dim oh(Q) :

Failure of the BRST Constru tionThe BRST onstru tion Q of modern string theory is ompletely unsu essful:sig(g oh(Q)) = 0 :

The string ontains hermitean zero modes fb0; 0g = 1 and therefore as manyphysi al states with positive norm as physi al states with negative norm.� = b0 0�+ 0b0�F1 = fb0� : � 2 FgF2 = f 0� : � 2 FghF1 F1i = 0 = hF2 F2iChoose a basis i in F1 and the dual basis �i in F2 ,

hi �ji = Æij ; hi +�i j +�ji = 2Æij = - hi -�i j -�ji :The subsidiary ondition b0phys = 0 pi ks zero norm states from F1 .

Stone-von Neumann TheoremAll unitary, irredu ible representations of[Xm; Pn℄ = i hÆmn ; [Xm; i h℄ = 0 = [Pn; i h℄ ; m;n 2 f0; 1; : : :D- 1g

are unitarily equivalent to the multipli ative and di�erential operations(Pn)(p) = pn(p) ; (Xm)(p) = -i��pm(p)

a ting on (a dense subset of) wavefun tions : R D ! C with s alar produ th� i = Z dDp��(p)(p) :

Hilbert spa e L2(R D): Set of equivalen e lasses of square integrable fun tions.� � , �(p) = (p) almost everywhere .

Stone-von Neumann TheoremAll unitary, irredu ible representations of[Xm; Pn℄ = i hÆmn ; [Xm; i h℄ = 0 = [Pn; i h℄ ; m;n 2 f0; 1; : : :D- 1g

are unitarily equivalent to the multipli ative and di�erential operations(Pn)(p) = pn(p) ; (Xm)(p) = -i��pm(p)

a ting on (a dense subset of) wavefun tions : R D ! C with s alar produ th� i = Z dDp��(p)(p) :

Ua = exp(-iamXm) , (Ua)(p) = (p- a) )No ompatible restri tion to fun tions whi h vanish in some domain (representa-tion is irredu ible).

No-state Theorem for the Covariant StringThe ovariant string and the toy relativisti quantum parti le employ[Xm; Pn℄ = i Æmn ; m;n 2 f0; 1; : : :D- 1g (1)

and restri t physi al states to mass shells (D dimensional volume = 0)��(P0)2 - ~P2�- 2(N- 1)�phys = 0where N has a dis rete spe trum.In D-dimensions phys(p) vanishes almost everywhere, phys � 0 .Ea h phys vanishes in a Hilbert spa e with a unitary representation of (1).A eptable quantum theories with a va uum do not ontain any Heisenberg pair[Xm; Pn℄ = iÆmn , but only momentum operators Pm su h as (P0)(p) =pm2 + p2(p) a ting on 2 L2(R D-1).

Negative Energy States of the Light one StringLight one String ontains [X-; P+℄ = i h a ting on 2 L2(R D-1) ; (p+;~p) 7! (p+;~p)multipli ative operator P- = 1P+�~P2 + 2(N- 1)� annot oexist with X-Hilbert spa e ontains with ea h also (exp(-iaX-))(p+;~p) = (p+-a;~p)i.e. also the negative energy shells

Ma key TheoremLmn ?= XmPn - XnPm - i 1Xk=1 aymk ank - aynk amkHow to onstru t the unitary representations of the Poin ar�e group withoutX0; X1; : : : XD-1?Ma key: Ea h unitary, irredu ible representation of the Poin ar�e group is unita-rily equivalent to the representation, whi h is indu ed by an irredu ible, unitaryrepresentation R of the little group of a ve tor p on its mass shellMm2 � R D.fp : p0 =pm2 + ~p2g � R D-1 , fp : j~pj =pjm2j+ (p0)2g � R�SD-2 , fp = 0g

:Mm2 ! C d ; h� i = Z dD-1pp0 ��T(~p)(~p) ; m2 � 0(U�)(p) = R(W(�;p))(�-1p)

Wigner RotationW(�2�1; p) =W(�2; �1p)W(�1; p)W(�;p) = L-1�p�Lp ; Lpp = pm2 > 0 ; p = (m; 0; 0; : : : 0) little group SO(D- 1)

Lp = p0m ~pTm~pm 1+ ~p~pTm(p0+m)!

in�nitesimal transformation (antihermitean)lij = -(pi ��pj - pj ��pi) + �ij ; �ij = -�ji = -(�ij)y[�ij; �kl℄ = Æik�jl - Æil�jk - Æjk�il + Æjk�ill0i = p0 ��pi + �ik pkp0 +m unique up to equivalen e

unlike George Jorjadze, Jan Plefka, Jonas Pollok: Bosoni String Quantization in the Stati Gauge

Massless RepresentationsW(�;p) = L-1�p�Lp ; Lpp = p

p0 =q(pz)2 + ~p2 ; p = (1; 1; 0; : : : 0) little group ISO(D- 2)Lp = DpBpBp =

0B�12(p0 + 1p0) 12(p0 - 1p0)12(p0 - 1p0) 12(p0 + 1p0) 1(D-2)�(D-2)1CA

Dp = �1 �Dp� ; �Dp = 0�pzp0 -~pTp0~pp0 1- ~p~pTp0(p0+pz)

1ADp shortest rotation from (p0; p0; 0 : : : ) to (p0; pz;~p),not de�ned for (p0; pz;~p) = (p0;-jpzj; 0)

In�nitesimal Transformations (antihermitean)lij = -(pi ��pj - pj ��pi) + ij ; i; j 2 f2; : : :D- 1g

[ ij; kl℄ = Æik jl - Æil jk - Æjk il + Æjk ill0i = p0 ��pi + ik pkp0 + pz ; p0 + pz = p+lzi = -(pz ��pj - pj ��pz) + ik pkp0 + pzloz = p0 ��pzSingular on the negative z-axis.Spin bundle with northern and southern hemisphere

Southern ChartLSp = DSpBSpDzxDzx rotation by � in zx-plane, Dzx(1; 1; 0 : : : )T = (1;-1; 0 : : : )TBSp =

0B� 12(p0 + 1p0) -12(p0 - 1p0)-12(p0 - 1p0) 12(p0 + 1p0) 1(D-2)�(D-2)1CA

DSp = �1 �DSp� ; �DSp = 0�-pzp0 ~pTp0- ~pp0 1- ~p~pTp0(p0-pz)

1ADSp shortest rotation from (p0;-p0; 0 : : : ) to (p0; pz;~p),not de�ned for (p0; pz;~p) = (p0; jpzj; 0) .

Southern In�nitesimal Transformationslij = -(pi ��pj - pj ��pi) + ij ; i; j 2 f2; : : :D- 1gl0i = p0 ��pi + ik pkp0 - pzlzi = -(pz ��pj - pj ��pz) - ik pkp0 - pzloz = p0 ��pzSingular on the positive z-axis .

Transition Fun tion

Tp=(LSp)-1LNp =0BBBB�1 1 -1+ 2p2xp2 2px~pTp2-2px~pp2 1- 2~p~pTp2

1CCCCA =0BBB�1 1 C S~nT-S~n 1+ (C- 1)~n~nT

1CCCAp2 = p2x + ~p2 (no p2z) , C = os 2' , S = sin 2' , p 6= (p0; pz; 0; : : : ; 0)Transition well de�ned also for half-integer Spin(D-2).The improved Light one string is a relativisti quantum theory, if the massiveSO(D- 2) modes onstitute SO(D- 1) multiplets.Known for D = 26, diÆ ulties with Young diagrams with more than 24 rows atN > 324?

Young Tables and FramesInhaltssumme 21 1 v1 2 t11;11;0Inhaltssumme 31 1 1 v1 1 2 t1 12 v2 3 t21;11;0 1;11;0

1

Young Tables and FramesInhaltssumme 41 1 1 1 v1 1 1 2 t1 1 12 v2 1 3 t2 2 2 v3 13 t2 4 t31;11;0 1;12;1 1;01;0

2

Young Tables and FramesInhaltssumme 51 1 1 1 1 v1 1 1 1 2 t1 1 1 12 v21 1 3 t2 1 2 2 v3 1 13 t2 1 22 v41 4 t3 2 3 t4 14 t4 23 v5 5 t51;11;0 1;12;1 2;12;0 2;11;0

3

Young Tables and FramesInhaltssumme 61 1 1 1 1 1 v1 1 1 1 1 2 t1 1 1 1 12 v21 1 1 3 t2 1 1 2 2 v3 1 1 13 t2 1 1 22 v4 1 12 2 v51 1 4 t3 1 2 3 t4 2 2 2 v6 1 14 t4 1 23 t5 1 32 v7 123 v81 5 t6 2 4 t7 3 3 v9 15 t7 24 t8 6 t91;11;0 1;12;1 2;1 1;1

3;1 3;1 1;13;1 2;01;04

Young Tables and FramesZerlegungen mit Inhaltssumme 7Erstzerlegungen1 1 1 1 1 1 1 v1 1 1 1 1 12 v2 1 1 12 2 v5 1 123 v9Unterdiagramme1 1 1 1 1 2 t1 1 1 1 1 3 t2 1 1 1 2 2 v3 1 1 1 13 t2 1 1 1 22 v41 1 1 4 t3 1 1 2 3 t4 1 2 2 2 v61 1 14 t4 1 1 23 t5 1 1 32 v7 1 2 22 v8 1 12 3 t51 1 5 t6 1 2 4 t7 1 3 3 t8 2 2 3 v101 15 t7 1 24 t8 1 33 t9 1 42 v11 2 23 v12 124 t91 6 t10 2 5 t11 3 4 t12 16 t11 25 t12 34 v13 7 t13

5

Young Tables and FramesRahmen 1;11;0 1;12;1 2;1 1;13;1 4;2 1;0 1;1

4;1 5;2 1;03;0 3;11;0

6

Young Tables and FramesZerlegungen mit Inhaltssumme 8Rahmen 1;11;0 1;12;1 2;1 1;13;1 4;2 2;1 1;1

5;2 6;2 3;2 2;15;1 7;2 2;14;1 3;01;0

7

Proof for all LevelsConje ture: At ea h level n, the polynomials P(ay) = k1m1::::::klmlaym1k1 : : : aymlkl withNP = nP , N =P1k=1PD-2i=1 k �ayik ��ayik - aik ��aik� transform under arepresentation of H =SO(D- 2), whi h is the restri tion of a representation ofG =SO(D- 1).Proof: UnknownClassi al Realizationlzi =X

n 6=0 �inn l-nj�0=0 ; flzi; lzjg = il0 1Xn=1 �

i-n�jn - �j-n�inn ; l0j�0=0 = 1Xn=1 �-n��n

i.e. lzipl0 and lij generate SO(D- 1) .

Covariant Stringlij = -(pi ��pj - pj ��pi) +Xk �ay ik ajk - ay jk aik�l0i = p0 ��pi -X

k �ay 0k aik - ay ik a0k�de�nes a Fo kspa e representation of the Lorentz algebra.The theorem of Del Guidi e, Di Ve hia and Fubini applies, that physi al statesin D = 26 have positive norm, though their proof assumes momentumeigenstates with dis rete s alar produ t.Their operators : exp(ikQ(z) : are de�ned only on states with k � p 2 Z andtherefore not on string state (wavefun tions) but only on auxiliary states, whi hare in ompatible with the Poin ar�e symmetry.Nevertheless, the DDF-proof of the positivity of the integrand in the s alarprodu t is orre t.

Ta hyoni Representation(U�)(p) = R(W(�;p))(�-1p)p = (0; : : : ; 0; 1), little group SO(1;D- 2) non ompa teither R = 1 or R in�nite dimensionalR = 1) ta hyon is a s alar, no supersymmetry, rep. an be worked outNo ommon onstru tion of the Poin ar�e transformations of the di�erent massshells.

Who ares?String amplitudes are not al ulated as matrix elements of string states. In theBRST formulation these states do not even onstitute a Hilbert spa e.Transition amplitudes of the S-matrix are derived from operators, whi h annotbe applied to string states but a t on a va uum. It is a no-string-state andde orates a subalgebra of vertex operators.