Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory

2.3 MB·

by  M. A. Escobar-Ruiz

Quantum and thermal fluctuations in quantum mechanics and field theories
from a new version of semiclassical theory
M.A. Escobar-Ruiz1,3,∗ E. Shuryak2,† and A.V. Turbiner1,2‡
1 Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico,
Apartado Postal 70-543, 04510 M´exico, D.F., M´exico
2 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA and
3 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
(Dated: May 20, 2016)
We develop a new semiclassical approach, which starts with the density matrix given by the
Euclidean time path integral with fixed coinciding endpoints, and proceed by identifying classical
(minimal Euclidean action) path, to be referred to as flucton, which passes through this endpoint.
Fluctuationsaroundfluctonpathareincluded,bystandardFeynmandiagrams,previouslydeveloped
forinstantons. WecalculatetheGreenfunctionandevaluatetheoneloopdeterminantbothbydirect
6
diagonalization of the fluctuation equation, and also via the trick with the Green functions. The
1
two-loop corrections are evaluated by explicit Feynman diagrams, and some curious cancellation
0
2 of logarithmic and polylog terms is observed. The results are fully consistent with large-distance
asymptotics obtained in quantum mechanics. Two classic examples – quartic double-well and sine-
y
Gordon potentials – are discussed in detail, while power-like potential and quartic anharmonic
a
oscillator are discussed in brief. Unlike other semiclassical methods, like WKB, we do not use the
M
Schro¨dinger equation, and all the steps generalize to multi-dimensional or quantum fields cases
straightforwardly.
9
1

I. INTRODUCTION wave functions do not have that, and basically are never
]
h used beyond say first and second orders.
-t Semiclassical approximations are well known tools, Of course, the higher level of generality comes with
p a heavy price. While classical part is relatively simple,
both in quantum mechanical and quantum field theory
e
already at one-loop level one needs to calculate determi-
h applications.
nants of certain differential operators. At two and more
[ Quantum mechanics itself originated from Bohr-
loops Feynman diagrams need to be evaluated on top
Sommerfeld quantization conditions, and semiclassical
3 of space-time dependent backgrounds: therefore those
v approximations for the wave function – the WKB and
can be done in space-time representation rather than in
4 its extensions – has been developed already in the early
energy-momentumonemostlyusedinQFTapplications.
6 daysofitsdevelopment, andaresinceastandardpartof
Most content of this paper is the explicit demonstration
9 quantum mechanics textbooks. Unfortunately, extend-
3 of how one can do all that, in analytic form, for two
ing such methods beyond one-dimensional cases or those
0 classic examples – quartic double-well and sine-Gordon
with separable variables proved to be difficult.
. potentials.
1
Semiclassical approximations in quantum field theory
0 Let us now outline briefly the history of semiclassi-
developeddifferently: theirstartingpointistheFeynman
6 cal evaluation of the path integrals in Euclidian time.
1 path integrals [1, 2], which is infinitely-dimensional any- Polyakov [5] used the example of symmetric double-well
: way,andthusthedimensionofquantummechanicscoor-
v potential to demonstrate the physical meaning of cel-
dinates or number of quantum fields is of secondary im-
i ebrated “instanton” solution in the non-Abelian gauge
X portance. So,theirmainadvantageoverthesemiclassical
theories (he and collaborators discovered shortly before
r approaches based on the Schr¨odinger equation (such as
that). For pedagogical presentation of this material, in-
a WKB) is that it can be used in multi-dimensional cases.
cluding the one-loop corrections, see [6]. Feynman dia-
Applications of such methods range from that by Rossi
grams and two-loop corrections have been calculated by
and Testa [3] in Quantum Field Theory (QFT) to recent
F. W¨ohler and E. Shuryak [7] for the double well po-
studies of protein folding [4] in statistical mechanics.
tential, extended to three-loops in our recent two papers
Another general advantage of the latter approach is [8,9]forboththedoublewellandsine-Gordonpotentials.
thatpathintegralsleadtosystematicperturbativeseries,
All the development was focused on the phenomenon
in the form of Feynman diagrams, with clear rules for
of quantum tunneling through the barrier for degenerate
each order. Text-book perturbative approaches for the
minima. Polyakov’s instanton is the classical path, com-
ingfromoneminimumofthepotentialtotheother. The
instanton amplitude, evaluated in the above mentioned
papers in higher orders, are approximations for the path
∗Electronicaddress: [email protected]
integral with the endpoints of the path corresponding to
†Electronicaddress: [email protected]
‡Electronic address: [email protected], alexan- thisarrangement,correspondingphysicallytoa“spectral
[email protected] gap”, the splitting between the lowest states of opposite
2
parity for the double well potential case. and the Green function. Since this correspondence has
From the theoretical point of view, the instanton am- neverbeenusedintheinstantonproblem,wediscussnon-
plitudes and perturbative series around them, are parts trivialsumrulefortheGreenfunctionfollowingfromthe
of more general construction nowadays known as trans- determinant value: as shown in Appendix B, the Green
series which include series at small coupling constant function used in our previous works has passed this test.
g: power-like terms ∼ gn, exponentially small terms InsectionVIIIweevaluatetwo-loopcorrectionsbydi-
∼ e−cogn2st, and logarithms of the coupling multiplied by rectevaluationofthediagramsovertheflucton,withsub-
such exponents, ∼ (logg)k e−cogn2st. The issue of unique tracted similar “vacuum diagrams”, fluctuations around
the trivial x(τ) = 0 vacuum. Surprisingly, all diagrams
definitionofsuchseriesisrelatedwiththesocalledresur-
yield analytic answers. While the individual diagrams
gencetheory,whichprovidecertainrelationsbetweense-
contain logs and polylogs, they all cancel in sum, lead-
ries near different extrema. Specific issues related to in-
ing to a rather simple analytic answer [25] . Would this
terplay between the perturbative series for trivial x = 0
property be true in two- and higher loop contributions:
path and instanton-antiinstanton contributions are ex-
it is interesting open question. Expansion of the results
tensively discussed e.g. in [10].
obtained for large displacement x is compared with the
Even more general question – whether these trans- 0
known asymptotic expansion of the ground state wave
series do define uniquely the whole function, represent-
function in Appendix A.
ing the path integral dependence on its parameters –
The final section X contains discussion of possible ap-
is the central item in rigorous mathematical definition
plicationstootherproblems,inquantummechanicswith
of the QFT’s. Related to it is the generalized defini-
severalvariables,statisticalmechanicsandquantumfield
tion of the path integrals, recently discussed by Witten
theories.
[11]. Noquestion,stillthereremainmanyopenquestions
related even with finite-dimensional integrals. Further-
more, even (1+0) dimensional path integrals – quantum
II. GENERAL SETTING
mechanical examples under consideration – still include
certain open theoretical problems which continue to at-
BydefinitiontheFeynmanpathintegralgivestheden-
tractattentionofphysicists(andmathematicians)today.
sity matrix in quantum mechanics [1]
In this paper we move from the well-trotted path of
tsutenandeloifnpgrtohbeaobriylitinyttooagosotmhreowuhgahttdhieffbearernriterd,irweecteivoanl.uaInte- ρ(x ,x ,t ) = (cid:90) x(ttot)=xf Dx(t)eiS[x(t)]/(cid:126) . (1)
i f tot
theprobability tofindaquantumsystematacertainpo- x(0)=xi
sition x inside classically forbidden region. It measures
0
HereS istheusualclassicalactionoftheproblem,e.g.
a “strength” of quantum effects, a quantum nature of
theproblem. Ingeneral,thisprobabilityisgivenbypath (cid:90) ttot (cid:20)m(cid:18)dx(cid:19)2 (cid:21)
integral in which the endpoints of the path coincide and S = dt −V(x) ,
2 dt
are fixed. We will develop a semiclassical theory for this 0
case. The corresponding classical solutions for it we will
for a particle of mass m in a static potential V(x) pro-
call fluctons, following the old paper of one of us [12]
videstheweightofthepathsin(1). Nowletusmovefrom
where it was introduced. Another early paper devoted
quantum mechanics to statistic mechanics, from quan-
to the subject was that by Rossi and Testa [3].
tum system to thermal system, from density matrix to
The paper is organized as follows. In section II the
probability. Step one is to rotate time into its Euclidean
general setting of the problem is explained, and the cor-
version τ = it. Step two is to define τ on a circle with
respondingclassicalsolutions,thefluctons,arederivedin
circumference β = τ . Such periodic time is known as
tot
thesectionIII.ThenextsectionIVtreatsquantumoscil-
the Matsubara time, and the density matrix of quantum
lations around classical path to quadratic order, result-
system is related to probability for thermal system with
ing in defining the corresponding determinant in section
temperature
V and the Green function in section VI.
Somewhat unexpectedly, we found that the quantum- T =(cid:126)/β . (2)
mechanical potential for fluctuations around flucton
backgroundinthedouble-wellproblemallowsexactana- Periodicityofthepathimpliesthatthereisonlyoneend
lyticsolutioninelementaryfunctions. Thereforewewere parameter x = x = x . The ensemble of such paths
i f 0
able to find analytic expression for the scattering phase represent equilibrium quantum statistical mechanics at
and evaluate the determinant via standard integral over temperature T, or, at T → 0, the ground state of the
its derivative. quantum system. See details of such setting in [2] and
Alternative derivation of the determinant is described many other sources on statistical field theory.
in section VII, in which its derivative over the coupling The main object we will be studying in this paper is
is related to certain Feynman diagram, which is evalu- the diagonal matrix elements of the density matrix in
ated using the (closed loop) Green function. Agreement coordinate representation, giving the probability for the
of those results shows consistency of the determinant specific coordinate value x (of the field φ ) to be found
0 0
3
in this ensemble. The basic expression for it we will use III. THE CLASSICAL PATHS: FLUCTONS
below is a path integral with endpoints fixed and coin-
cided Forpedagogicalreasons,wewillproceedusingparticu-
larexamples,forwhichexpressionscanbesimpleenough
P(x ,β)=(cid:90) x(β)=x0Dx(τ)e−SE[x(τ)]/(cid:126) . (3) to allow analytic evaluation of all quantities. The main
0
ideaisthatinEuclideantimetheeffectivepotentialflips
x(0)=x0
and the classical minimum becomes a maximum. There-
thus, we will consider all (closed) trajectories starting fore classical paths with E=0 “slipping down” from a
and ending at x , where S = (cid:82)βdτ[m(dx)2+V(x)]. maximum to any point exist.
0 E 0 2 dτ (I).Itishardnottostartwiththeharmonicoscillator,
There are two basic limits of this expression (3). One is
asthefirstexample. Onecanalwaysselectunitsinwhich
at large β, or low T. Using standard definition of the
the particle mass m = 1 and the oscillator frequency
density matrix in terms of states with definite energy
ω =1, so that our Lagrangian is written as
(cid:88)
P(x ,β)= |ψ (x )|2e−Enβ , (4) x˙(τ)2 x(τ)2
0 n 0
L = + . (7)
E
n 2 2
Note that, for positivity, the Euclidean sign change we
oneseesthatthislimitP correspondstothelowest–the
applynottothekinetic, buttothepotentialterm. Any-
ground state
way,intimeτ theoscillatordoesnotoscillatebutrelaxes,
P(x ,β →∞)∼|ψ (x )|2 (5) theclassicalequationofmotion(EOM)producesolutions
0 0 0 of the kind eτ,e−τ. The flucton solution at E = 0 on a
circle with circumference β can be easily found as their
Intheoppositecaseofsmallβ thecircleissmallandone
superposition satisfying
can ignore time dependence of the paths. In this limit
x(0) = x(β) = x , (8)
0
P(x0,β)∼e−V(Tx0) , (6) namely,
corresponding to classical thermal distribution in a po- (cid:0)eβ−τ + eτ(cid:1)
x (τ) = x . (9)
tential V. Needless to say, the expression is correct for flucton 0 eβ +1
any T.
defined for τ ∈ [0,β]. At low T (or large β) it is con-
venient, due to periodicity in τ, to shift its range to
2.0
τ ∈ [−β/2,β/2]. At zero T = 1/β the range becomes
infinitely large, and solution becomes simply x e−|τ|. At
0
1.5
highT,ontheotherhand,the“thermalcircle”getssmall
β →0, it can be just approximated by x .
1.0 0
The classical action of such a path is
0.5 (cid:18)β(cid:19)
S = x2 tanh , (10)
flucton 0 2
0.0
(cid:45)4 (cid:45)2 0 2 4
it tells us that the particle distribution
12
(cid:32) (cid:33)
10 x2
P(x )∼exp − 0 , (11)
8 0 coth(β)
2
6 is Gaussian at any temperature. Note furthermore, that
4 the width of the distribution
(cid:18) (cid:19)
2 <x2 > = 1coth β = 1 + 1 , (12)
2 2 2 eβ −1
0
(cid:45)4 (cid:45)2 0 2 4
canberecognizedasthegroundstateenergyplusonedue
to thermal excitation. These results are, of course, very
FIG. 1: Time dependence of the classical flucton solution
wellknown,seee.g. Feynman’sStatisticalMechanics[2].
y (τ),see(25)(upperplot)andthecorrespondingpoten-
fluct
tial(1+W),see(37)ofthefluctuations(lowerplot),bothfor (II). Our next example is the symmetric power-like
x =2,λ=0.1. potential
0
g2
V = x2N , N =1,2,3,... , (13)
2
4
for which we discuss only the zero temperature β = transitionbetweenthemhappenswhentheinstantonac-
1/T → ∞ case. The (Euclidean) classical equation at tion S[x (τ)] = 1/12λ is larger or smaller than one,
inst
zero energy x˙2 =V(x) has the following solution respectively.
2
Standardstepsareselectingunitsforηsuchthatω =1
x0 and shifting the coordinate by it,
x (τ)= , x >0,(14)
fluct (cid:0)1 + g(N −1)xN−1|τ|(cid:1)N−1 0
0
x(τ)=y(τ)+η , (23)
with the action
so that the potential (21) takes the form
2gxN+1
S[xfluct] = N +0 1 , (15) y(τ)2 (cid:16) √ (cid:17)2
V = 1+ 2λy(τ) , (24)
2
hence
corresponding to harmonic oscillator well at small y.
(cid:32) (cid:33)
P(x )∼exp −2gxN0 +1 , (16) The flucton solution, the minimal action path for the
0
N +1 path integral (3), in which the path is forced to pass
through the point x at τ =0 now takes the form
0
whichisinacompleteagreementwithWKBasymptotics
x
at x0 →∞ [26]. yfluct(τ)= √ 0 √ , (25)
(III). The third example is the anharmonic oscillator e|τ|(1+ 2λ x0)− 2λ x0
potential
We remind that in zero T case, or infinite circle β →∞,
1 τ ∈ (−∞,∞), and solution exponentially decreases to
V = x2 (1+gx2) , g >0 , (17)
2 both infinities, see Fig.1. Its generalization to finite T is
straightforward.
atzerotemperatureβ =1/T →∞. Theclassicalflucton The action of this solution is
solution with the energy E =0 is given by √
x (τ) = √gx0 , (18) S[yfluct]=x20(1+ 2 23λx0) , (26)
fluct (cid:112)
cosh(τ)+ 1+gx2 sinh(τ)
0
and thus in the leading semiclassical approximation the
which leads to the flucton action probability to find the particle at x takes the form
0
S[x0] = 23 (1+gxg20)32 −1 . (19) P(x0)∼exp(cid:32)−x20− 2√32λx30(cid:33) (27)
In the limit g →0 we recover the action of the harmonic
In the weak coupling limit only the first term remains,
oscillator and at x →∞ we obtain
0
corresponding to Gaussian ground state wave function
√
2 g 1 2 1 of the harmonic oscillator. In the strong coupling limit
S[x (τ)] = x3+ √ x − +O( ) . (20)
fluct 3 0 g 0 3g x the second term is dominant, and the distribution then
0
corresponds to well known cubic dependence on the co-
in complete agreement with the asymptotic expansion of ordinate. These classical-order results are of course the
the ground state wave function squared (see Appendix same as one gets from a standard WKB approximation.
A) [27]. When |x0| < η the classical flucton solution can be
However, for the most detailed studies we select two constructed from the pieces of the instanton and antiin-
other examples. stantonsolutions. Inthisregion,fluctons,instantonsand
(IV). One is the quartic one-dimensional potential antiinstantons are distinct classical paths, all contribut-
ing to the path integral (3).
V(x) = λ(x2−η2)2 , (21)
(V). Our last example is the sine-Gordon potential
withtwodegenerateminima. Tunnelingbetweenthemis
described by well known instanton solution 1
V = (1−cos(gx)) , (28)
g2
1
x (τ) = η tanh( ω(τ −τ )) , (22)
inst c
2 with infinite number of degenerate vacua. Tunneling be-
tween adjacent vacua is described by well known instan-
assuming that ω2 = 8λη2. Note that the instanton has
ton solution
arbitrary time location τ , while the flucton does not.
c
We will discuss both the weak coupling limit of small 4
x (τ) = arctan(eτ) . (29)
λ, and the strong coupling limit of large λ. In fact, the inst g
5
Inthezerotemperaturecase,orverylargecircleβ →∞, witheigenvalue 3 ,aswellasthecontinuumofunbound
4ω2
the flucton solution has a very simple form states with eigenvalue above ω2. Since one has the ana-
lytic expression for the scattering phase δ , the determi-
(cid:20) (cid:21) p
4arccot eτ cot(gx0) nant has been evaluated so to say “by definition”, using
4
complete set of states, for a review see e.g. [6]. A new
x (τ) = . (30)
fluct
g relationbetweenthedeterminantandtheGreenfunction
for the instanton we will discuss in section B.
The classical action for this solution is In the case of flucton classical solution (25) the poten-
16 sin2(gx0) tial of the fluctuations we put into the form
S[x ]= 4 , (31)
fluct g2 V(cid:48)(cid:48)(y )=1+W ,
fluct
and,thus,intheleadingsemiclassicalapproximationthe
where
probability to find the particle at x takes the form
0
6X(1+X)e|τ|
P(x )∼exp[−16 sin2(g4x0)] . (32) W = (e|τ|−X+Xe|τ|)2 . (37)
0 g2
The classical path depends on 3 parameters of the prob-
lem, λ,x and ω (which we already put to 1): but in W
0
IV. FLUCTUATIONS AROUND THE the first two appear in one combination only
CLASSICAL PATH √
X ≡ x 2λ . (38)
0
Nowweturntoquantumfluctuationsaroundtheclas-
This observation will be important in Section V. An ex-
sical path
ampleof(1+W)isshowninFig.1(lowerplot). Notethat
y(τ)=y (τ)+f(τ) , (33) W exponentially decreases at large τ.
fluct
In the sine-Gordon case the potential of the fluctua-
which is absent in the instanton case. Let us put this tions has the following form
expression into the action and expand it to the needed
1
orderinf. Butbeforewedoso, letusremindthereader V(cid:48)(cid:48)(y )=
that, bythedefinition, allpathsshouldpassthroughthe fluct (1+e−2τtan2(X˜))2
same point at τ = 0 and, thus, there is an important
condition
[1+e−4τtan4(X˜)−6e−2τtan2(X˜)] , τ >0 , (39)
f(0)=0 . (34)
where the relevant combination of parameters is
Since the classical path is a local minimum of the ac-
gx
tion,thereforethereisnotermO(f1). Smallfluctuations X˜ ≡ 0 , (40)
4
are described by the Lagrangian
cf (38).
f˙(τ)2 f(τ)2
L= +V(cid:48)(cid:48)(y ) + O(f3) , (35)
fluct
2 2
V. THE FLUCTON DETERMINANT
where we used a short hand notations V(cid:48)(cid:48)(y ) =
fluct
∂2V(y)/∂y2| . Its variation leads to Schr¨odinger-
like equationy=wyitfhluctthe potential V(cid:48)(cid:48). The operator governing quadratic fluctuations around
ForharmonicoscillatorthispotentialV(cid:48)(cid:48) isjustacon- flucton is
stant, so in this case the fluctuations do not depend on Of ≡ −f¨(τ)+V(cid:48)(cid:48)(y )f(τ) , (41)
the classical path. Higher order derivatives of V all van- fluct
ish, hence, in this case all fluctuations are just Gaussian.
where the derivative has already been described above
For quartic double-well potential for the famous clas- (37). At large |τ| the nontrivial part of the potential
sical solution x (t) (22), the instanton, the potential
inst disappears and solutions have a generic form
entering has the well known form
ψ (τ)∼sin(pτ +δ ) , (42)
(cid:18) (cid:19) p p
3
V(cid:48)(cid:48)(y )=ω2 1− . (36)
fluct 2cosh2(ωτ/2) where for momentum p, only the scattering phase δp de-
pends on the potential. The eigenvalues of the operator
This potential is one of few exactly solvable quantum O are, for the double well example (37), simply,
mechanical problems. There are two bound states, the
famouszeromodewitheigenvaluezeroandanotherstate λ =1+p2 , (43)
p
6
and the determinant DetO is their infinite product. Its At large L and n one can replace summation to an inte-
logarithm is the sum gral, resulting in the generic expression
log DetO = (cid:88)log(1+p2) , (44) logDetO = (cid:88)log(1+p2)
n n
n n
where the sum is taken over all states satisfying zero
boundary condition on the boundary of some large box. (cid:90) ∞ dpdδ
Taking the path integral over fluctuations around the = p log(1+p2) . (47)
π dp
classical path, in the Gaussian approximation, leads to 0
the following standard expression
After using few different numerical methods for par-
ticular values of the parameter X, we discovered that
exp(−S[x ])
P(x )= flucton there exist exact (non-normalized) analytic solution for
0 (cid:112)
Det(Oflucton) the eigenfunctions of the form
ψ (τ) ∼ sin(pτ +∆(p,τ)) F(p,τ) , (48)
×[1+O(two and more loops)] , (45) p
with the following two functions
withO =O definedin(41). Inthissectionwedis-
flucton
cuss numerical evaluation of the determinant: another (cid:20) (cid:21)
−3p(1+2X)
method will be discussed in the section V, after we will ∆(p,τ)=arctan
1−2p2+6X+6X2
derive the corresponding Green function for the fluctu-
ations in section VI. Calculation of two and more loop
corrections via Feynman diagrams will be discussed in (cid:20) (cid:21)
N
section VIII. + arctan ,
As it is well known, the nontrivial part of the problem D
is not in the eigenvalues themselves, but in the counting
where
of levels. Standard method (see e.g. § 77 of [14]) vanish-
ing boundary conditions at the boundary of some large
box, at τ =L, leads to
N = 3p[1+2X+X2−X2e−2τ] ,
p L+δ =πn , n=1,2,... . (46)
n pn
D = (2p2−1)(1+X2)−2e−τ(cid:0)2(1+p2)−e−τ(2p2−1)(cid:1)X+(2p2−1)e−2τ −4e−τ(1+p2) ,
(cid:20) (cid:18) (cid:19)
1
F(p,τ) = × e4τ(1+5p2+4p4)+4e3τ(1+p2) 2−4p2+eτ(1+4p2) X+
(eτ −X+eτX)2
(cid:18) (cid:19) (cid:18)
6e2τ 3+p2+4p4+4eτ(1−p2−2p4)+e2τ(1+5p2+4p4) X2+4eτ 2(1−p2−2p4)+
(cid:19) (cid:18)
6e2τ(1−p2−2p4)+3eτ(3+p2+4p4)+e3τ(1+5p2+4p4) X3+ 1+5p2+4p4+
(cid:19) (cid:21)1/2
8eτ(1−p2−2p4)+8e3τ(1−p2−2p4)+6e2τ(3+p2+4p4)+e4τ(1+5p2+4p4) X4 .
It is important that at τ =0 the solution (48) goes to zero: according to the flucton definition, all fluctuations at
this point must vanish (34). It is the condition which fixes the scattering phase.
At large time, where all terms with decreasing exponents in ∆(p,τ) disappear and the remaining constant terms
define the scattering phase, we need
(cid:20) (cid:21) (cid:20) (cid:21)
3p(1+2X) 3p
δ =arctan −arctan . (49)
p 1−2p2+6X+6X2 1−2p2
Comments: (i) the scattering phase is O(p) at small p;
7
(ii) it is O(1/p) at large p and, thus, there must be a Since the calculation above includes only a half of the
maximum at some p; time line, τ > 0, and the other half is symmetric, the
(iii) for X = 0 two terms in (49) cancel out. This needs completeresultforthelogDetOshouldbedoubled. Sub-
to be the case since in this limit the nontrivial potential stituting (49) to (47) we obtain a (surprisingly simple)
W of the operator also disappears; exact result
(iv) at large time the amplitude F (48) goes to a con-
stant, as it should.
The arctan-function provides an angle, defined modulo Det(O) = (1+X)4 . (50)
the period, and thus it experiences jumps by π. For-
tunately, its derivative dδ /dp entering the determinant
p
(47) is single-valued and smooth. The momentum de- NotethatatX =0wereturntoharmonicoscillatorcase.
pendence of the integrand of this expression for X = 4 VI. THE GREEN FUNCTION OF THE
is shown in Fig.2(a). Analytic evaluation of the integral FLUCTUATIONS AROUND THE FLUCTON
(47)wasnotsuccessful,theresultsofthenumericaleval- SOLUTION
uation are shown by points in Fig.2(b). However, the
guess 2log(1+X), shown by the curve in Fig.2(b) hap-
pens to be accurate to numerical accuracy, and thus it General procedure for inversion of the operator (41),
must be correct. We will demonstrate that it is exact leadingtoaGreenfunction, isdifferentfortheinstanton
below. and flucton cases. In the instanton case the inversion is
only possible in the subspace normal to the zero mode,
0.6 leading to specific difficulties. The flucton problem we
discuss now has no shift symmetry (no translation in-
0.4
variance) and thus no zero modes. Needless to say that
thissymmetryiskilledbytheboundaryconditionatthe
0.2
fixed moment, f(τ =0)=0.
0.0
The corresponding equation to be solved thus is
-0.2
-0.4 ∂2G(τ ,τ )
− 1 2 +V(cid:48)(cid:48)(y (τ ))G(τ ,τ ) (51)
-0.6 ∂τ2 fluct 1 1 2
1
0 5 10 15 20
10
8
= δ(τ −τ ) .
1 2
6
4
Homogeneous equation (with zero r.h.s.) has two so-
lutions
2
0
0 20 40 60 80 100
eτ
f (τ) = , (52)
FIG.2: (a)Theintegrandof(47),log(1+p2)dδ /dp,versusp, 0 (eτ(1+X)−X)2
p
for X =4. (b) The integral (47) vs parameter X: points are
numerical evaluation, line is defined in the text.
and
e−τ (cid:18) (cid:19)
f (τ) = 8X3(1+X)eτ +12X2(1+X)2τe2τ −8X(1+X)3e3τ +(1+X)4e4τ −X4 (53)
1 2(X−eτ(1+X))2
(Hereafter we only discuss half line τ > 0). The first solution – would be zero mode if shift be allowed – is
exponentiallydecreasingatlargetime,thesecondoneisincreasingintime. Standardconstructionimmediatelyyields
8
the following Green function
G(τ1, τ2) = (cid:0) e−|(cid:1)τ12−(cid:0)τ2| (cid:1)2(cid:20)8 e21(τ1+τ2+3|τ1−τ2|)X3(1+X)
2 eτ1(1+X)−X eτ2(1+X)−X
− 8 e12(3τ1+3τ2+|τ1−τ2|)X(1+X)3+ e2(τ1+τ2)(1+X)4−6e(τ1+τ2+|τ1−τ2|)X2(1+X)2|τ1−τ2|
(cid:18) (cid:19)
+ e(τ1+τ2+|τ1−τ2|) 6X4(τ +τ ) + 12X3(1+τ +τ )+ 6X2(3+τ +τ )+4X−1
1 2 1 2 1 2
(cid:21)
−e2|τ1−τ2|X4 , (54)
for τ , τ >0 .
1 2
Similarly, in the sine-Gordon problem the same standard construction yields the following Green function
1 1
G(τ , τ ) = × ×
1 2 8(cid:0)cosh(τ )+cos(2X˜) sinh(τ )(cid:1) (cosh(τ )+cos(2X˜) sinh(τ ))
1 1 2 2
(cid:20)
1
2(τ +τ −|τ −τ |) sin2(2X˜)+ 8 cos(2X˜) sinh2( (τ +τ −|τ −τ |))
1 2 2 1 1 2 2 1
2
(cid:21)
+ (3+cos(4X˜)) sinh(τ +τ −|τ −τ |) , (55)
1 2 2 1
for τ , τ >0 . is a perturbation: its effect can be evaluated by the fol-
1 2
lowing Feynman diagram
VII. RELATING THE DETERMINANT AND
THE GREEN FUNCTION
(cid:90)
∂logDet(O ) ∂V (τ)
flucton flucton
= dτG(τ,τ) ,(57)
∂X ∂X
The method we will use in this section relies on the
following observation. When the fluctuation potential
depends on some parameter, it can be varied. In the
case at hand (37), the potential we write as containingderivativeofthepotentialasavertexandthe
“loop”, the same point Green function, see Fig.3. This
V =1+W(X,τ) ,
flucton relates the determinant and the Green function [28] : if
the r.h.s. of it can be calculated, the derivative over X
depends on the combination (38). Its variation resulting
can be integrated back.
in extra potential
In the quartic double-well problem the Green function
∂W
δV = δX (56) loop propagator is
flucton
∂X
1
G(τ, τ) = (58)
(cid:0) (cid:1)4
2 X−eτ(1+X)
(cid:18) (cid:19)
× −X4+8eτX3(1+X)−8e3τX(1+X)3+e4τ(1+X)4+e2τ(−1+4X+18X2+12X3+12X2(1+X)2τ) ,
and the “vertex” With these expressions one can evaluate the r.h.s. of
∂V (τ) 6eτ(cid:0)X+eτ(1+X)(cid:1)
flucton
= . (59)
∂X (cid:0)−X+eτ(1+X)(cid:1)3
9
the relation (57), and adding the same expression for
negative time, one gets the result
∂logDet(O ) 4
flucton
�V = , (60)
G(⌧, ⌧) ∂X 1+X
which exactly agrees with the result (50) from the direct
evaluation of the determinant using the phase shift. So,
theGreenfunctionhaspassedaverynontrivialtest,and
we conclude that it is ready to be used for evaluation of
FIG. 3: Symbolic one-loop diagram, including variation of two and higher loop diagrams.
the fluctuation potential δVand the simplified “single-loop”
Green function G(τ,τ). In the sine-Gordon problem the corresponding simpli-
fied expression for G(τ,τ) is
(cid:20)
G(τ, τ) = 1 × 4(e2τ −1)2cos(2X˜)−cos(4X˜) (61)
(cid:18) (cid:19)2
4 1+e2τ +(e2τ −1)cos(2X˜)
(cid:21)
+ e4τ(cid:0)3+cos(4X˜)(cid:1)+ 8e2ττ sin2(2X˜)−3 .
Evaluating the one-loop diagram Fig.3, we arrive at the result
log Det(O ) = 4 tan[X˜] . (62)
flucton
For the power-like potential (13) the “Green function loop” takes the form
−1+(N −1)X1τ +(1+X1(1−N)τ)12−NN
G(τ, τ) = , (63)
(3N −1)X
1
where
X =gxN−1 ,
1 0
and the “vertex” reads
∂V (τ) 2(2N −1)NX
flucton 1
= , (64)
∂X ((N −1)τX −1)3
1 1
(τ <0). Hence, we obtain the result
2N
log Det(O ) = logX .
flucton N −1 1
In the case of the anharmonic oscillator (17) the “Green function loop” is
(cid:20)
G(τ, τ) = (sinh(τ)+cosh(τ)X2) −6τX (sinh(τ)+cosh(τ)X )(cid:0)−1+X2(cid:1) (65)
4X (cosh(τ)+sinh(τ)X )4 2 2 2
2 2
(cid:18) (cid:19)(cid:21)
+sinh(τ) 4+X [sinh(2τ)+3(−3+cosh(2τ))X +3sinh(2τ)X2+(5+cosh(2τ))X3] ,
2 2 2 2
where while the “vertex” is given by
(cid:113)
X = 1+gx2 , ∂Vflucton(τ) 12(sinh(τ)+cosh(τ)X2)
2 0 = , (66)
∂X (cosh(τ)+sinh(τ)X )3
2 2
10
(τ >0). Thus, where B = B (X). Like in the calculations near the
n n
instantonsolution,weneedtoseparatethefiniteflucton-
log Det(Oflucton) = 2log[X2(1+X2)] . related contribution for each diagram from the infinite
(time-divergent)contributionwithoutit. Thisisdoneby
subtracting the same expression with “vacuum vertices”
VIII. HIGHER ORDER FEYNMAN DIAGRAMS
√
v = 6 2λ , (69)
3,0
Now, using only the tools from quantum field theory,
the Feynman diagrams in the flucton background, we
compute the two-loop correction to the density matrix
v = 24λ , (70)
(45)forthedouble-wellpotential. Inprinciple,thehigher 4,0
order diagrams are evaluated by standard rules.
Unlike the calculations near the instanton solution and the “vacuum propagator”
[7, 8], in the case of flucton there are no zero modes
and related Jacobian, so all diagrams follow from the e−|τ1−τ2| e−τ1−τ2
Lagrangian. In the quartic double-well potential, the G0 =G(τ1, τ2)|X→0 = 2 − 2 . (71)
flucton-based Green function (54) was determined above
and the only vertices are triple and quartic ones
(Notethat(71)differsfromthevacuumpropagatorinthe
√
6 2λ(X+eτ(1+X)) instanton problem where the second term in the r.h.s. is
v3(τ) = −X+eτ(1+X) , (67) absent. In particular, it is no longer translational invari-
ant because of extra boundary condition at τ = τ = 0
1 2
for fluctuations at the fixed point.)
v4 = 24λ . (68) Thetwo-loopcorrectionB1 weareinterestedincanbe
written as the sum of three diagrams, see Fig.4, diagram
The loop corrections in (45) are written in the form
a which is a one-dimensional integral and diagrams b
1
and b corresponding to two-dimensional ones.
∞ 2
(cid:88) 1
[1+O(two and more loops)] = 2 B λn, B = , Explicitly, we have
n 0
2
n=0
1 (cid:90) ∞ 3
a ≡ − v [G2(τ, τ)−G2(τ, τ)]dτ = (72)
8λ 4 0 560X2(1+X)4
0
(cid:18)
× 24X−60X2−520X3−1024X4−832X5−245X6+24(1+X)2(1+2X)(−1+6X(1+X))log(1+X)
(cid:20) (cid:21)(cid:19)
X
+288X2(1+X)4PolyLog 2, ,
1+X
here PolyLog[n,z]=(cid:80)∞ zk/kn is the polylogarithm function and
k=1
1 (cid:90) ∞(cid:90) ∞
b ≡ [v (τ )v (τ )G3(τ , τ )− v v G3(τ , τ )]dτ dτ (73)
1 12λ 3 1 3 2 1 2 3,0 3,0 0 1 2 1 2
0 0
(cid:18)
1
= × −24X+60X2+520X3+1024X4+832X5+245X6
280X2(1+X)4
(cid:20) (cid:21)(cid:19)
+24(1+X)2(cid:0)1−4X−18X2−12X3(cid:1)log(1+X)−288X2(1+X)4PolyLog 2, X ,
1+X
1 (cid:90) ∞(cid:90) ∞(cid:2) (cid:3)
b ≡ v (τ )v (τ )G(τ , τ )G(τ , τ )G(τ , τ )− v v G (τ , τ )G (τ , τ )G (τ , τ ) dτ dτ (74)
2 3 1 3 2 1 1 1 2 2 2 3,0 3,0 0 1 1 0 1 2 0 2 2 1 2
8λ
0 0