- Email: [email protected]
Construction of physical states in Yang-Mills theory: The time-like gauge
Volume
72B, number
3
PHYSICS
CONSTRUCTION
OF PHYSICAL
LETTERS
STATES
THE TIME-LIKE
2 Janudry
IN YANG-MILLS
1978
THEORY
GAUGE*
P SENJANOVIC 1 Department
of Physzcs, Umverszty of Cahforma, Berkeley,
CA 94720,
and TheoretIcal Physzcs Group, Bans Kzdrrch Instrtute, 11001 Belgrade,
Received
13 October
We construct physical states m pure Yang-Mills theory basis m the physical subspace of Hllbert space Comparison
Recent developments m pseudopartlcle physics (for an excellent review, see ref [ 11) have revived mterest m the time-like gauge Ai = 0 A heurlstlc construction of the Hllbert space of physlcal states m this gauge has been performed recently by Eylon [2] Eylon represents a physical state by an mfimte set of gauge-equlvalent configurations Motivated by Eylon’s results, we have asked ourselves the following question does there exist a simple procedure for constructmg the whole physical subspace of the total Hllbert space m the usual approach This is clearly a very important question. irrespectively of Its relation to Eylon’s method As we shall see m this paper, the answer to our query will be m the affnmatlve Moreover, we shall show that our method yields results Identical to Eylon’s The time-like gauge YanggMdls theory possesses the Hamlltoman
The n’s are momenta
conjugate
1977
m the time-hke gauge AZ = 0 We also construct IS made with a recent paper by Eylon
* Research ’
a complete
tentials Estabhshmg the validity of correct operator equatlons of motion will lustlfy the canonical formahsm introduced above This 1s readily performed (4)
(5) which, upon combmmg eqs (4) and (5), IS seen to be simply the time-hke gauge version of a portion of the usual Lagranglan equations of motion The remammg portion IS recovered m the physical subspace of Hllbert space (by physical states we shall mean states that are invariant under gauge transformations which depend on space coordmates only) smce that portlon IS simply the generator of such mfmlteslmal gauge transformation, so when applied on a physical state It must vamsh The system defined by eqs (1) to (3) IS a pretty conventional quantum field-theoretic system, so by standard methods of quantum mechamcs we can estabhsh the existence of a complete basis {la)} m the correspondmg Hllbert space (all equations hold at fixed time)
to A: A;(x)la)
They commute
USA
Yugoslavia
among themselves, as do the vector po-
sponsored by the US National Science Found&on under Grant No 74-08175-A02 Present address Department of Theoretical Physics, Oxford Umverslty, Oxford, Great Britain
= LI;(x)~L’,
(6)
s Da/a) (ai = 1, (QlQ’)=h[Q-Q’]
(7) =
Define now the state
v
6[Q(X)-Q’(X)]
(8)
la4)
329
Volume
72B, number
AP(x J
3
PHYSICS
= (ug)qx>la, J ’
(9)
where (a”$0 1s the gauge transform x”(Lz”),” = s$?;
A@ sg +
(1/e)
s;
g
LETTERS
P2 = JBgfihDaDblag)
of ap
+&=O
(11,12)
Xp are the group generators, and g(x) are parameter functions describing the gauge group Eq (12) guarantees we stay m the time-like gauge Obviously lag) also has the followmg property
(qg 4lag) = u1%2~),
=
(10)
(13)
s
Dg~hDblb%
s[a - bh] (bl
(19)
(bl
As the group mtegratlon 1s mvarlant under group multlphcatlon, we can write the followmg formal expression P2 = JDgDhDblbg)
(bl = (Jl)h)P=
nflG,P x
(20)
The mfmlte constant m eq (20) will reflect itself m the non-normahzablhty of physlcal states P IS Hermltean
Pt = JDgDa/a)
where
1978
(a/Z+) (bl
= “DgiShDaDblag) J
J
sg,
2 January
(agl =JDgDog-’
lag-‘) (aI
= s ~~DeIng-l)(al=~ijlg-lDolag-l)(al =
s DgDalag)
(21)
(ai = P -1
Moreover, with Ug defined by eq (14), V$z) = lag)
(15)
Using the property (15) and eq (17), we fmd the following useful form for Ug Ug = JDnlug)
(al
(16)
Finally, we define a proJection operator
P=J@Ug
=&Daiag)
(al
We have used the Identities Dug = Da and Dg-’ = Dg, the first of which 1s trivially established, and the second of which 1s well known We now claim that 9$,
= P9(,
where 81,, 1s the physical subspace and BI IS the full Hllbert space The proof is simple If l+)=J$h
(17)
In the functional integral over all possible g(x) m eq (17), we mean by Dg
(22)
*Ugl$)=
=JDgl$)=(I-J
IJ/,‘Pl$)=
SDgCJgI@
dq.)l$w~FPBI*~ph~PB(
If
(18) where M(g(x)) 1s the local group measure, and fi2, IS the total group volume (we consider compact groups only) This peculiar normahzatlon, as will be seen later, ~11 lead us to the same normahzatlon of physical states as Eylon’s P ISnot a projector m a usual mathematical sense, but IS close to that, as w,e show below
330
= Ug j-DaDhl& = s DaDhI& ~
(01~) =JDai%l~?~)
(alx)
(all) = P/x)
= Iti>* I$)GXph
*P9Kslp,,
~ofmallyP81~81~~,qed Hence the query we posed has been answered m the affirmative We would now like to compare our method
Volume
72B, number
3
PHYSICS
to Eylon’s Corresponding have a physical state ldph = Plu)
to a conflguratlon
lAph =PIa%=jUgI&
,,(biSl”),h = (rj
ah(x), with h(x) arbitrary,
=jDglu”)=ldph
(24)
d&WMx))l
ph (’ la’ph
=
GlPtPlQ)
= (blP%
(25)
=( ~~~)(hl~i~)=SD~(b,~g)=S~gslhg-o]
=
ph(biSia)p,, = QlPSPIa)
J
Dg @%Sleg”)
fl&Doia)ph
ph(ai =p,
(31)
where /Da 1s a functional integral over gauge-mequlvalent configurations only, defined by
(26) SDa being an ordinary functional of eq (3 1) is as follows
We now use two ingredients pug0 = P,
integral
The proof
(27) P=JiSgDajug)
PSP=PS
(blPSl&la)
Using now the standard canonical path integral corresponding to the operator formalism defined by eqs (1). (3), and observmg that, upon integration over the canonical momenta, it yields Eylon’s Lagranglan, we can see that we end up with Eylon’s “Assumption 3” the uutlal conflguratlon IS alzy configuration m the set (ag, all g} (Eylon’s c~), and one sums over all configurations m the set {bg, all g} (Eylon’s al)*’ Needless to say, while our construction mlmlcks Eylon’s m terms of its properties, the philosophy IS different Eylon postulates the physical Hllbert space, while we project it out of an extended Hllbert space The question arises as to the construction of a complete basis m the physical subspace We show now that the basis ja)ph is complete in % ph (y
which IS the same normahzatlon as Eylon’s How about the comparison with Eylon’s S-matrix prescrlptlon? One obtains
Ji&)
(30)
(23)
Thus, an infinite set of gauge-equivalent configurations IS associated with a single physical state This mlmlcks Eylon’s postulate that “a physical state IS represented by an mfimte set of gauge-equivalent conflguratlons” The normahzatlon of states m eq (24) IS [we call Dg = n,
1978
a(x) we
=jDglug)
Hence for the confIguration we have the physical state
2 January
LETTERS
l-I &=iG x
The first ingredient 1s trivial after what we learned To prove the second one, we only need remember properties of group integration and the fact that the S operator 1s gauge invariant Hence
= j-L)hDgUh
(~1
=JDgDhDaiahg)
(ah 1,
(33)
(28)
Ug-l S= JDhDgUhg-l
= jkhDgl$S= We can therefore write
PS n &=& x
where we have used eq (32) Next, changing variables hg +g, D(hg) = Dg, we fmd P=JUUJL)~~~~)JD~GZ’/II
fiG, X
(34)
which IS recognized to be the same as eq (3 l), upon using eq (23) While our paper was motivated by Eylon’s, and hence was presented m a manner designed to reveal that motlvatlon, the reader will no doubt recognize that our method has a lot of merit (and power) on its
S
(29) *-1Once we have converged
to Eylon’s result, we can apply the rest of his analysis to study the physical S matrix m more detail and clarify Its properties
331
Volume
72B, number
3
PHYSICS
own For Instance, we can construct any sort of physlcal state by applying our proJectlon operator, while Eylon only has states that are generahzatlons of states which dlagonahze vector potentials We shall display fully the power of our method m a future pubhcatlon, m which we Intend, on the basis of our prelnmnary investlgatlons, to clarify the followmg questions 1 How to construct physical sectors with .9 # 0 (ours, like Eylon’s have 0 = 0) 2 How to construct physical coherent states In the same pubhcatlon, we shall also deal with the questlon of construction of simple states I$) that obey Mandelstam’s crlterlon for the finiteness of energy density m the axial gauge [3] G(x,, x2)/$) = 0, where G(xl, x2) IS the generator of residual gauge transformations We shall do this by usmg an adaptation of methods we Introduced in the present letter We are grateful1 to Y Eylon, E Glldener, A Jevlckl, S Mandelstam, A Patrasclolu, and E Rabmovlcl for
332
LETTERS
2 January
1978
discussions We thank the Aspen Center for Physics, where this paper was completed, for hospltallty Note added After the completion of the present paper we discovered a section in the paper by Bernard and Weinberg [4], where the idea behind our paper IS hinted at, and which has some overlap \nth Eylon’s paper
References [ 1] W Marclano and H Pagels, Rockefeller report COO2232B-3 30 [2] Y Eylon, LBL report LBL-6476 (1977) [3] S Mandelstam, mvlted talk at the Washmgton meetmg of the American PhysIcal Society (1977) [4] Bernard and E Wemberg, Phys Rev D (June 15, 1977)
72B, number
3
PHYSICS
CONSTRUCTION
OF PHYSICAL
LETTERS
STATES
THE TIME-LIKE
2 Janudry
IN YANG-MILLS
1978
THEORY
GAUGE*
P SENJANOVIC 1 Department
of Physzcs, Umverszty of Cahforma, Berkeley,
CA 94720,
and TheoretIcal Physzcs Group, Bans Kzdrrch Instrtute, 11001 Belgrade,
Received
13 October
We construct physical states m pure Yang-Mills theory basis m the physical subspace of Hllbert space Comparison
Recent developments m pseudopartlcle physics (for an excellent review, see ref [ 11) have revived mterest m the time-like gauge Ai = 0 A heurlstlc construction of the Hllbert space of physlcal states m this gauge has been performed recently by Eylon [2] Eylon represents a physical state by an mfimte set of gauge-equlvalent configurations Motivated by Eylon’s results, we have asked ourselves the following question does there exist a simple procedure for constructmg the whole physical subspace of the total Hllbert space m the usual approach This is clearly a very important question. irrespectively of Its relation to Eylon’s method As we shall see m this paper, the answer to our query will be m the affnmatlve Moreover, we shall show that our method yields results Identical to Eylon’s The time-like gauge YanggMdls theory possesses the Hamlltoman
The n’s are momenta
conjugate
1977
m the time-hke gauge AZ = 0 We also construct IS made with a recent paper by Eylon
* Research ’
a complete
tentials Estabhshmg the validity of correct operator equatlons of motion will lustlfy the canonical formahsm introduced above This 1s readily performed (4)
(5) which, upon combmmg eqs (4) and (5), IS seen to be simply the time-hke gauge version of a portion of the usual Lagranglan equations of motion The remammg portion IS recovered m the physical subspace of Hllbert space (by physical states we shall mean states that are invariant under gauge transformations which depend on space coordmates only) smce that portlon IS simply the generator of such mfmlteslmal gauge transformation, so when applied on a physical state It must vamsh The system defined by eqs (1) to (3) IS a pretty conventional quantum field-theoretic system, so by standard methods of quantum mechamcs we can estabhsh the existence of a complete basis {la)} m the correspondmg Hllbert space (all equations hold at fixed time)
to A: A;(x)la)
They commute
USA
Yugoslavia
among themselves, as do the vector po-
sponsored by the US National Science Found&on under Grant No 74-08175-A02 Present address Department of Theoretical Physics, Oxford Umverslty, Oxford, Great Britain
= LI;(x)~L’,
(6)
s Da/a) (ai = 1, (QlQ’)=h[Q-Q’]
(7) =
Define now the state
v
6[Q(X)-Q’(X)]
(8)
la4)
329
Volume
72B, number
AP(x J
3
PHYSICS
= (ug)qx>la, J ’
(9)
where (a”$0 1s the gauge transform x”(Lz”),” = s$?;
A@ sg +
(1/e)
s;
g
LETTERS
P2 = JBgfihDaDblag)
of ap
+&=O
(11,12)
Xp are the group generators, and g(x) are parameter functions describing the gauge group Eq (12) guarantees we stay m the time-like gauge Obviously lag) also has the followmg property
(qg 4lag) = u1%2~),
=
(10)
(13)
s
Dg~hDblb%
s[a - bh] (bl
(19)
(bl
As the group mtegratlon 1s mvarlant under group multlphcatlon, we can write the followmg formal expression P2 = JDgDhDblbg)
(bl = (Jl)h)P=
nflG,P x
(20)
The mfmlte constant m eq (20) will reflect itself m the non-normahzablhty of physlcal states P IS Hermltean
Pt = JDgDa/a)
where
1978
(a/Z+) (bl
= “DgiShDaDblag) J
J
sg,
2 January
(agl =JDgDog-’
lag-‘) (aI
= s ~~DeIng-l)(al=~ijlg-lDolag-l)(al =
s DgDalag)
(21)
(ai = P -1
Moreover, with Ug defined by eq (14), V$z) = lag)
(15)
Using the property (15) and eq (17), we fmd the following useful form for Ug Ug = JDnlug)
(al
(16)
Finally, we define a proJection operator
P=J@Ug
=&Daiag)
(al
We have used the Identities Dug = Da and Dg-’ = Dg, the first of which 1s trivially established, and the second of which 1s well known We now claim that 9$,
= P9(,
where 81,, 1s the physical subspace and BI IS the full Hllbert space The proof is simple If l+)=J$h
(17)
In the functional integral over all possible g(x) m eq (17), we mean by Dg
(22)
*Ugl$)=
=JDgl$)=(I-J
IJ/,‘Pl$)=
SDgCJgI@
dq.)l$w~FPBI*~ph~PB(
If
(18) where M(g(x)) 1s the local group measure, and fi2, IS the total group volume (we consider compact groups only) This peculiar normahzatlon, as will be seen later, ~11 lead us to the same normahzatlon of physical states as Eylon’s P ISnot a projector m a usual mathematical sense, but IS close to that, as w,e show below
330
= Ug j-DaDhl& = s DaDhI& ~
(01~) =JDai%l~?~)
(alx)
(all) = P/x)
= Iti>* I$)GXph
*P9Kslp,,
~ofmallyP81~81~~,qed Hence the query we posed has been answered m the affirmative We would now like to compare our method
Volume
72B, number
3
PHYSICS
to Eylon’s Corresponding have a physical state ldph = Plu)
to a conflguratlon
lAph =PIa%=jUgI&
,,(biSl”),h = (rj
ah(x), with h(x) arbitrary,
=jDglu”)=ldph
(24)
d&WMx))l
ph (’ la’ph
=
GlPtPlQ)
= (blP%
(25)
=( ~~~)(hl~i~)=SD~(b,~g)=S~gslhg-o]
=
ph(biSia)p,, = QlPSPIa)
J
Dg @%Sleg”)
fl&Doia)ph
ph(ai =p,
(31)
where /Da 1s a functional integral over gauge-mequlvalent configurations only, defined by
(26) SDa being an ordinary functional of eq (3 1) is as follows
We now use two ingredients pug0 = P,
integral
The proof
(27) P=JiSgDajug)
PSP=PS
(blPSl&la)
Using now the standard canonical path integral corresponding to the operator formalism defined by eqs (1). (3), and observmg that, upon integration over the canonical momenta, it yields Eylon’s Lagranglan, we can see that we end up with Eylon’s “Assumption 3” the uutlal conflguratlon IS alzy configuration m the set (ag, all g} (Eylon’s c~), and one sums over all configurations m the set {bg, all g} (Eylon’s al)*’ Needless to say, while our construction mlmlcks Eylon’s m terms of its properties, the philosophy IS different Eylon postulates the physical Hllbert space, while we project it out of an extended Hllbert space The question arises as to the construction of a complete basis m the physical subspace We show now that the basis ja)ph is complete in % ph (y
which IS the same normahzatlon as Eylon’s How about the comparison with Eylon’s S-matrix prescrlptlon? One obtains
Ji&)
(30)
(23)
Thus, an infinite set of gauge-equivalent configurations IS associated with a single physical state This mlmlcks Eylon’s postulate that “a physical state IS represented by an mfimte set of gauge-equivalent conflguratlons” The normahzatlon of states m eq (24) IS [we call Dg = n,
1978
a(x) we
=jDglug)
Hence for the confIguration we have the physical state
2 January
LETTERS
l-I &=iG x
The first ingredient 1s trivial after what we learned To prove the second one, we only need remember properties of group integration and the fact that the S operator 1s gauge invariant Hence
= j-L)hDgUh
(~1
=JDgDhDaiahg)
(ah 1,
(33)
(28)
Ug-l S= JDhDgUhg-l
= jkhDgl$S= We can therefore write
PS n &=& x
where we have used eq (32) Next, changing variables hg +g, D(hg) = Dg, we fmd P=JUUJL)~~~~)JD~GZ’/II
fiG, X
(34)
which IS recognized to be the same as eq (3 l), upon using eq (23) While our paper was motivated by Eylon’s, and hence was presented m a manner designed to reveal that motlvatlon, the reader will no doubt recognize that our method has a lot of merit (and power) on its
S
(29) *-1Once we have converged
to Eylon’s result, we can apply the rest of his analysis to study the physical S matrix m more detail and clarify Its properties
331
Volume
72B, number
3
PHYSICS
own For Instance, we can construct any sort of physlcal state by applying our proJectlon operator, while Eylon only has states that are generahzatlons of states which dlagonahze vector potentials We shall display fully the power of our method m a future pubhcatlon, m which we Intend, on the basis of our prelnmnary investlgatlons, to clarify the followmg questions 1 How to construct physical sectors with .9 # 0 (ours, like Eylon’s have 0 = 0) 2 How to construct physical coherent states In the same pubhcatlon, we shall also deal with the questlon of construction of simple states I$) that obey Mandelstam’s crlterlon for the finiteness of energy density m the axial gauge [3] G(x,, x2)/$) = 0, where G(xl, x2) IS the generator of residual gauge transformations We shall do this by usmg an adaptation of methods we Introduced in the present letter We are grateful1 to Y Eylon, E Glldener, A Jevlckl, S Mandelstam, A Patrasclolu, and E Rabmovlcl for
332
LETTERS
2 January
1978
discussions We thank the Aspen Center for Physics, where this paper was completed, for hospltallty Note added After the completion of the present paper we discovered a section in the paper by Bernard and Weinberg [4], where the idea behind our paper IS hinted at, and which has some overlap \nth Eylon’s paper
References [ 1] W Marclano and H Pagels, Rockefeller report COO2232B-3 30 [2] Y Eylon, LBL report LBL-6476 (1977) [3] S Mandelstam, mvlted talk at the Washmgton meetmg of the American PhysIcal Society (1977) [4] Bernard and E Wemberg, Phys Rev D (June 15, 1977)