Table Of ContentZero Modes in Light-Front φ4
1+1
G. B. Pivovarov∗
Institute for Nuclear Research,
Moscow, 117312 Russia
(Dated: January 4, 2005)
Within a scheme of light front quantization of φ4 , it is demonstrated that dynamics of zero
1+1
modesimpliesphasetransition, andthatthecriticalvalueofthecouplingcoincideswiththeoneof
theequal time quantization.
PACSnumbers: 11.10.Ef,11.10.Kk
There is a discussion in the literature on how phase
transitionsarerelizedwithin the lightfrontquantization
5 dk
0 of field theories (a discussion of this issue and relevant φ(x)=Z 4πω (k) a(k)e−ikx+a†(k)eikx . (2)
0 references see in [1]). It is demonstrated in this note lf (cid:2) (cid:3)
p
2 thatdynamicsofzeromodescanbecompletelydecoupled Here x is a shorthand for x− (x0 x1)/√2, and
n from the rest of the dynamics, and implies the presence ω (k)= k2+(m2+ǫk2)(ǫ+4≡k2/M2)−.
a of phase transitions. The zero modes are the modes of lf
In termps of the above creation annihilation operators,
J zerolight-likemomentum. Inourconsideration,weusea
the Hamiltonian is
0 regularizationunderwhichthemodesofzeromomentum
1 are replaced with modes of low momenta (soft modes). H = dk ν(k)a†(k)a(k)+ g dxφ4(x), (3)
Momentaofsoftmodesvanishwhentheregularizationis Z 4Z
1
removed.
v where the energy of excitation is
Theregularizationweusewasintroducedin[1]. Itisa
4
6 regularizationintroducedon the levelofthe Lagrangian. ν(k)= ωlf(k)−k . (4)
0 The regularization involves higher derivatives to regu- ǫ+4k2/M2
1 larize the ultraviolet divergencies, and a small external
IntheHamiltonian,thenormalproductoffieldsisim-
0 tensor coupledto derivativesofthe field under quantiza-
5 plied in φ4. Perturbatively, this Hamiltonian leads to
tion. TheexternaltensorbreakesLorentzinvariance,and
0 Feynman diagramsthat have finite limits at the regular-
is needed to avoid infinities caused by infinite volume of
/ ization removed. These limits coincide with the expres-
h the Lorentzgroup. Inparticular,regularizedLagrangian
-t of φ41+1 is stiioonn,swobitthaintheediwntitehgirnattihonesstinantdhaerdlolnigghittudfrionnatl qmuoamnteinztaa-
p
e restricted to positive values.
1 1 1
h = gµνφ φ + ǫtµνφ φ + ((cid:3)φ)2 V(φ), (1) The purpose of this note is to consider the limit of
: Lr 2 ,µ ,ν 2 ,µ ,ν 2M2 − regularizationremoved nonperturbatively.
v
Xi where (cid:3)φ=gµνφ,µν, and V(φ)=m2φ2/2+gφ4/4. The gaTuswsioanobesffeercvtaitvieonpsowteinlltilaeladreloautredcotnosiHdearmatilitoonn.iaFnir(s3t),
regularization is removed in the limit ǫ 0,M .
ar Thetensortµν isasymmetrictensorwhos→enonzero→co∞m- coincides in the limit of regularizationremoved with the
one computed for φ4 within equal time quantization,
ponents in a Lorentz frame are t++ =1,t−− = 1. The 1+1
− andimpliesthepresenceofphasetransition[2]. Asnoted
role of the term with the t-tensor is to break Lorentz
in [2], modes with momentum of order √ǫ are crucial in
invariance. The signs of the components of this tensor
the derivation. Second, if we neglect the modes of low
arechosentohaveanonnegativelight-frontHamiltonian
momentum, and the states with light front energy go-
(see [1]).
ing to infinity in the limit of regularizationremoved,the
StandardquantizationcanbeappliedtotheaboveLa-
Hamiltonian (3) is equivalent to the conventional light
grangian, if x+ (x0 +x1)/√2 is taken as dynamical
≡ frontHamiltonian(fortheconventionallightfrontHamil-
time, and the initial conditions for the field are set on a
tonian, see [3] and Eq. (6) below). Let us detail the
light front at fixed x+.
second observation: At fixed small or negative k, excita-
This quantization leads to a Hamiltonian, the opera-
tion energy ν(k) goes to infinity when the regularization
tor, representing the P+ compnent of the total momen- is removed. Neglecting these infinite energy excitations,
tum of the system. As explained in [1], it is convenient
andreplacingω (k) with its limit (ω(k) k), we arrive
lf
toexpresstheHamiltonianintermsofthecreationanni- →
at the conventionallight frontdecomposition of the field
hilationoperatorsrelatedtothefieldoperatorasfollows:
in creation annihilation operators, and the conventional
light front Hamiltonian:
∞ dk
φ (x)= a(k)e−ikx+a†(k)eikx , (5)
∗Electronicaddress: [email protected] lf Z0 √4πk (cid:2) (cid:3)
2
Straightforward consideration leads to the following
expressions for H and H :
∞ m2 g s int
H = dk a†(k)a(k)+ dxφ4 (x). (6)
lf Z 2k 4Z lf 1
0 H = [H P ], (12)
s et et
√ǫ −
In the last formula, we have taken into account that, at
fixedpositivek,ν(k) m2/2k inthelimitofregulariza-
→
tion removed. 3g
We concludethat Hamiltonian(3) canbe decomposed Hint =gΦs(0)Z dxφ3lf(x)+ 2 Φ2s(0)Z dxφ2lf(x).
as follows:
(13)
Here H and P are respectively the Hamiltonian and
H =H +H +H , (7) et et
s int lf the momentum of φ4 in equal time quantization:
1+1
where H is the contribution of the soft modes with mo-
s g
mentum oforder √ǫ, and Hint is the contributionto the Het =Z dk ω(k)b†(k)b(k)+ 4Z dxΦ4s(x), (14)
Hamiltonian describing the interaction between the soft
and conventional modes. Both H and H are negli-
s int
gible within perturbation theory, but are important for
understanding of the phase transition. P = dk kb†(k)b(k). (15)
et Z
To derive H and H , we split the momentum axis
s int
into two domains, soft and normal. The soft momenta
(Therelationν(√ǫk) (ω(k) k)/√ǫshouldbeusedin
are restricted in magnitude with a cut Λc that scales to → −
the derivation of Eq. (12), where ν(k) is defined in Eq.
zero as ǫ 0, but slower than √ǫ. In this way, the nor-
→ (4).)
mal modes occupy the whole momentum space at regu-
WeconcludethatHamiltonian(3)containsapiecethat
larizationremoved,and,atthesametime,thesoftmodes
scalesas1/√ǫ. Thus,thispiececangiveafinitecontribu-
alwayscontainmomentaoforder√ǫ. Forthesoftmodes,
tiontotheenergy,ifH P isoforder√ǫ,orzero. For-
we rescale the momenta dividing them by √ǫ. Techni- et et
−
tunately, we know that H defines a theory with a mass
cally,thiscanbedoneviareplacingcreationannihilation et
gap(ifthe couplingdoes notequalthe criticalvalue)[4].
operators as follows:
Thus H P can be of order √ǫ only if P is of order
et et et
−
a(k)= 1 b( k ), a†(k)= 1 b†( k ). (8) 1/√ǫ, which is impossible, because of the cutoff Λc. We
ǫ41 √ǫ ǫ41 √ǫ concludethattheonlywaytokeeptheenergyfinite isto
restrict the state space of the soft modes to the vacum
Notice that the scale of b, b† is such that they obey the vectorofHet,inwhichcasethecontributionproportional
canonical commutation relations, as a, a† do (explicitly, to Het Pet vanishes.
[b(k),b†(p)] = δ(k p)). Our prescription for studying After−this conclusion, we see that Φs(0) and Φs(0)2 in
−
thelimitǫ 0istoreplacea-operatorswithb-operators Eq. (13) should be replaced with their vacuum expec-
→
eachtime the momentumofa-operatoris lowerthanΛ . tations. Ultimately, after the regularization is removed,
c
In doing this replacement, the following object is of use:
3g
H =H +gΦ dxφ3 (x)+ Φ2 dxφ2 (x), (16)
dk lf Z lf 2 Z lf
Φ (x) [b(k)e−ikx+b†(k)eikx], (9)
s ≡Z 4πω(k)
whereΦ andΦ2 arethe vacuumexpectationsofthe field
p
whereω(k)=√k2+m2. Notice,thatthisfieldisdecom- operator and field operator squared (with normal prod-
uct assumed) of φ4 . At small coupling, both vacuum
posed into b-operators in the way taking place in equal 1+1
expectations are zero, and H = H . As we know from
time quantization. The last prerequisite we need for de- lf
the properties of H , there is a criticalvalue of the cou-
riving H and H is the relationbetween a piece of the et
s int
pling beyond which the expectations are nonzero.
field involving the soft modes,
Let us summarise our findings. Dynamics of the soft
Λc dk modesisidenticaltothedynamicsofthecompetetheory.
φs(x)≡Z−Λc 4πωlf(k) [a(k)e−ikx+a†(k)eikx], (10) ITthies doneclyoutpralecdeforfomthethperedsyennacmeiocfstohfethsoeftnomrmodaelsminodtehse.
p
reguralizedtheory that survivesthe removalofthe regu-
and Φ (x) of Eq. (9). It is easy to establish in the limit
s
larization is the phase transition. At coupling exceeding
ǫ 0 the following relation:
→ the critical value, new terms should be introduced into
φ (x)=Φ (√ǫx). (11) the light front Hamiltonian to account for nonzero vac-
s s
uum expectations of the soft modes.
(The relations ω (ǫk) √ǫω(k), Λ /√ǫ should This work was supported in part by RFBR grant no.
lf c
→ → ∞
be used.) 03-02-17047.
3
[1] V.T.Kim,G. B.Pivovarovand J.P.Vary,Phys.Rev.D 301, 299 (1998).
69, 085008 (2004) [arXiv:hep-th/0310216]. [4] J. Glimm and A. Jaffe, Quantum Physics. A Func-
[2] G. B. Pivovarov, Gaussian Effective Action in Light tionalIntegralPointOfView (Springer-Verlag,NewYork,
Front φ4 (To appear in proceedings of QUARKS-2004) 1981).
1+1
[arXiv:hep-th/0501051]
[3] S. J. Brodsky, H. C. Pauli, and S. S. Pinsky, Phys. Rept.
