Unique Translation between Hamiltonian Operators and Functional Integrals

4 Pages·0.12 MB·English

by  T. Gollisch

Unique Translation between Hamiltonian Operators and Functional Integrals
Tim Gollisch∗ and Christof Wetterich†
Institut fu¨r Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
(February 1, 2008)
A careful treatment of the discretization errors in the path integral formulation of quantum
mechanics leads to a unique prescription for the translation from the Hamiltonian to the action in
thefunctionalintegral. Anexampleisgivenbyan interaction quadraticin theoccupation number,
characteristicformanybodybosonicsystems. Asaresult,thetermlinearintheoccupationnumber
(chemical potential) receives a correction as compared to the usual formulation based on coherent
states. A perturbativecalculation supports therelevance of this correction.
05.30.Jp, 67.40.Kh, 03.65.Db
1
0
0
2 I. INTRODUCTION Usually, µ˜ is interpreted as the chemical potential µ and
v˜(q) as the interaction potential v(q) [3,9,10].
n
a Many modern techniques in particle and statistical The intuitive derivation makes use of the coherent
J physicsrelyontheoriesgivenintheformoffunctionalin- states introduced by Glauber [11] and can be found in

4 tegrals,while the basicdescriptionofthe systemis often Ref. [3]. It has been suggested earlier, though, that
1 easier and more intuitively formulated on the operator the discretization errors, namely the (∆τ)2–corrections

level. The transition between these two languages has in the exponent, are of importance and cannot be dis-
2
been given much attention (see e.g. [1–8]) and is recog- carded thoughtlessly [12]. In this work, we present an
v
3 nizedasamathematicallycomplicatedissueiftheHamil- approachwhich controls these higher order corrections.
7 tonian cannot be separated into a momentum and a lo- As a consequence, the parameter µ˜ obtains a correc-
2 cation dependent part. Nevertheless, calculations based tion from the mathematical manipulations and is to be
8
on functional integral formulations have been very suc- interpretedasµ+Ω,whereΩisaconstantdependingon
0
0 cessful in applications, for instance in the context of the v(q). The derivation we present here leads to a unique
0 renormalizationgroup in statistical physics. relationshipbetweenthe Hamiltonianandthe functional
/ In order to interpret the parameters which appear in integral.
t
a the functional integral correctly, it is necessary to con-
This issue is related to the work of Christ and Lee
m
nect them to the parameters of the Hamiltonian opera-
[6] who obtain extra terms in the quantum Yang–Mills
- tor . This translation between the parameters in the
d H Hamiltonian from a gauge transformation on the oper-
functionalintegralandtheHamiltonianhastobeunique
n ator level. They claim that this is connected to non–
o sincephysicalquantitiescanbecomputedbothways. (If negligible terms of higher order in the discretization pa-
c necessary,manybodymodels haveto be regularizedsuch
rameter ǫ in the functional integral. The existence of
v: thatthe expectationvaluesofH andother operatorsare such possible subtleties in the discretizationis our start-
i welldefined. Typically,thiscanbedonebydiscretization
X ing point. We propose a formulation that avoids those
on a lattice.)
unpleasant higher order contributions.
r
For an ensemble of interacting bosons of mass m with
a There has been recent significant improvement in the
the Hamiltonian
understanding of phase space integrals by de Boer,
q2 1 Peeters,Skenderis,andvanNieuwenhuizen[7,8],seealso
= µ a†a + a† a† v(q )a a , (1)
H 2m− q q 2 q1−q3 q2+q3 3 q1 q2 [13]. They derive the action and Feynman rules for
Xq (cid:18) (cid:19) q1X,q2,q3 Hamiltonian operators in curved spacetime which are at
the partition function is written as a functional integral most quadratic in pˆ. This is applied to the non–linear
over a complex field ϕ (τ) sigma model. In this work we are interested in the sta-
q
tistical physics applications offunctional integrals where
Z = ϕ exp [ϕ,ϕ∗] (2) the interactions typically involve higher powers of pˆ. We
D {−S } therefore take a different approach, which allows us to
Z
treat momentum and location operators on equal foot-
with the action
ing. (There is no restriction to Hamiltonians at most
q2 quadratic in pˆ). Our simple treatment leads (for flat
[ϕ,ϕ∗] = dτ ϕ∗ µ˜+∂ ϕ
S q 2m − τ q spacetime)toaperhapsunexpectedresult,namelyacor-
Z hXq (cid:18) (cid:19) rectionintheactionwhichwecanconnecttoexperimen-
1
+ ϕ∗ ϕ∗ v˜(q )ϕ ϕ . (3) tal findings such as the critical chemical potential at the
2 q1−q3 q2+q3 3 q1 q2 lambda–transition of helium–4.
q1X,q2,q3 i
1
II. FUNCTIONAL INTEGRAL IN QUANTUM + x p p x x p p x
k+2 k+1 k+1 k+1 k+1 k k A k
h | ih | ih | ih |H | i
MECHANICS
+ x p p x x p p x
k+2 B k+1 k+1 k+1 k+1 k k k
h |H | ih | ih | ih | i
+ x p p x x p p x
k+2 k+1 k+1 k+1 k+1 B k k k
The mathematical issues of path integration are most h | ih | ih |H | ih | i
easily studied in one dimensional quantum mechanics, +2ǫ2 xk+2 pk+1 pk+1 A xk+1 xk+1 pk pk (cid:1)A xk
h | ih |H | ih | ih |H | i
whichis afinite theory. The generalizationofthe results +x p p x x p p x
(cid:0)k+2 B k+1 k+1 k+1 k+1 B k k k
h |H | ih | ih |H | ih | i
from this toy model to quantum field theory is straight-
+x p p x x p p x
k+2 k+1 k+1 A k+1 k+1 B k k k
forwardandgivenbelow(IV). Weconsiderthecanonical h | ih |H | ih |H | ih | i
+x p p x x p p x .
partition function with the Hamiltonian h k+2|HB| k+1ih k+1|HA| k+1ih k+1| kih k| ki
(7)
(cid:1)(cid:3)
λ
=ma†a+ a†a†aa. (4)
H 2
Intheaboveexpression,allmatrixelementsaretrivialto
We introduce location and momentum operators xˆ = evaluate, and we see that the ǫ2–term is almost exactly
a†+a /√2 and pˆ = i a† a /√2, which obey the the one we need for reconverting the expansion into an
−
usual commutation relation [xˆ,pˆ] = i. Bringing the in- exponential. All we need to do is adjust some of the in-
(cid:0) (cid:1) (cid:0) (cid:1)
teraction term into a symmetric ordering yields dices (k+2 k+1 k) in this term, but this gives a
↔ ↔
correctionof (ǫ3) (as will be evident in (9)). Hence we
m 3λ O
= + + (5a) can write
A B
H H H − 2 8
with dx0 dx2N−1 dp0 dp2N−1
Z = ... ...
N
√2π √2π √2π √2π
1 λ Z
HA = 4(m−λ) pˆ2+xˆ2 + 16 xˆ4+2pˆ2xˆ2+pˆ4 , (5b) 2N−1 ǫ
exp ip (x x ) (m λ) x2+p2
HB = 41(m−λ)(cid:0)pˆ2+xˆ2(cid:1)+ 1λ6(cid:0)xˆ4+2xˆ2pˆ2+pˆ4(cid:1). (5c) × (Xk=0h k k+1− k − 2 − (cid:0) k k(cid:1)
ǫλ
The term −21λ xˆ(cid:0)2+pˆ2 (cid:1)as we(cid:0)ll as the const(cid:1)ant term − 8 x4k+p2kx2k+x2k+1p2k+p4k +O(ǫ3)). (8)
m + 3λ arise from the use of the commutators in the (cid:0) (cid:1)i
−2 8 (cid:0) (cid:1)
manipulation of (4). The trivial constant term will be
neglected for notational simplicity. The crucial question is whether ZN has a well defined
Thisparticularorderingisasuitablestartingpointfor limit for N such that the corrections (ǫ3) can
→ ∞ O
the formulation of the functional integral. We will show be neglected. Since this is closely related to the issue of
thatitavoidsunpleasantdiscretizationcorrectionswhich rapidlyvaryingxandp(asfunctionsofτ =βk/(2N)),it
wouldbepresentforotherformulationsliketheonebased is useful to perform a Fourier transform. The definition
on coherent states. of the Matsubara–modes x˜n, p˜n
We denote the locationand momentumeigenstates by
x and p and follow the standard procedure for writ- N
|ingi the p|airtition function as a functional integral (with x = e2πink/(2N)x˜ , (9a)
k n
ǫ≡β/(2N) and x2N ≡x0): n=X−N
N
Z =Tre−βH =ZN pk = e2πin(k+1/2)/(2N)p˜n (9b)
N−1 n=−N
= dx x e−2ǫH x ... x e−2ǫH x . X
2a 2N 2N−2 2 0
h | | i h | | i
Z(cid:18) a=0 (cid:19)
Y (with x˜∗ = x˜ , p˜∗ = p˜ and x˜ , p˜ , x˜ , p˜ real)
(6) n −n n −n 0 0 N N
corresponds in the imaginary time language to taking
In expanding the exponentials for small ǫ, we will take the variables p(τ) at locations between the x(τ). From
specialcareoftheǫ2–contributions. Thematrixelements (9), we see that shifting the k–index introduces a fac-
are evaluated by inserting further location and momen- tor of 1 + (1/N). We also turn to the more conve-
O
tum eigenstates nient language of complex fields by substituting x˜n =
ϕ∗ +ϕ /√2andp˜ =i ϕ∗ ϕ /√2.Thisyields
x 1 2ǫ +2ǫ2 2 x −n n n −n− n
k+2 k
h | − H H | i (cid:0) (cid:1) (cid:0) (cid:1)
= dpk+1 dxk+1 dpk ZN = ϕ exp ( +∆ )+ (1/N2) (10)
D − S S O
Z Z
x p p x x p p x (cid:2) (cid:3)
k+2 k+1 k+1 k+1 k+1 k k k
× h | ih | ih | ih | i
ǫ(cid:2) xk+2 pk+1 pk+1 A xk+1 xk+1 pk pk xk with
− h | ih |H | ih | ih | i
(cid:0)
2
= [2πinϕ∗ϕ +β(m λ)ϕ∗ϕ ] quantum mechanics. The two conclusions we draw from
S n n − n n
n thiswillbethatthefunctionalintegralgivesfiniteresults
X
βλ and therefore needs no regularization and that only the
+ ϕ∗ ϕ∗ ϕ ϕ , (11)
2 n1−n n2+n n1 n2 quadratic term m λ gives correct quantitative results.
n1X,n2,n We start with t−he simple quantum mechanical calcu-
πn
∆ = 4iNsin 2πin ϕ∗ϕ lation. We denote the eigenstates of the unperturbed
S 2N − n n
Xn h (cid:16) (cid:17) i Hamiltonian operator (λ = 0) by |li. With hl|pˆ2|li =
βλ π(n +n ) l+ 1, we obtain the thermal expectation value
+ 1 cos 1 2 (ϕ∗ ϕ ) 2
16 − 2N −n1+n− n1−n
n1X,n2,n(cid:20) (cid:18) (cid:19)(cid:21) ∞ l e−βml−12βλl(l−1) 1
×(ϕ−∗n2−n−ϕn2+n)(ϕ−∗n2 −ϕn2)(ϕ−∗n1 −ϕn1). (12) hpˆ2i= Pl=∞l=00 e−βml−12βλl(l−1) + 2. (14)
We notice that the coupling constants in the ∆ – Due to the fastPconvergence of the two sums, this can
S
correction are (1/N2). (The n–dependence is not
easily be evaluated numerically.
∝ O
important in this respect. Up to a constant prefactor of
In the functional integral, the expectation values of
Z,the contributionsof the highn Matsubara–modesare
operators are easily computed by adding to source
effectively cut off by suppression factors n−2 from the terms like Kpˆ2, e.g. pˆ2 = ∂lnZ(K)/∂K .HAs long
squared propagator, which arise from th∝e first term in h i K=0
as the source terms do not involve products of pˆand xˆ,
.) Eq. (10) is therefore particularly suited for an un- (cid:12)
they simply add to the action the corresp(cid:12)onding terms
S
equivocaldefinitionofthefunctionalintegralasthelimit with the replacements f(pˆ) dτ f i(ϕ∗ ϕ)/√2
N istaken. Weextendthen–summationfrom → −
→∞ −∞ and f(xˆ) dτ f (ϕ∗+ϕ)/√2 . One obtains
to in the action and drop ∆ as well as the other → R (cid:0) (cid:1)
∞ S S pˆ2 = 1 (ϕ∗ ϕ)2 with the usual definitions of ex-
(1/N2)–corrections. h i −2h −R i (cid:0) (cid:1)
O pectation values in the functional integral O(ϕ,ϕ∗) =
In the “imaginary time language” (using ϕ(τ) = h i
Z−1 ϕ O(ϕ,ϕ∗)exp( ). A standard calculation in
ne2πinτ/βϕn), the action corresponds to first orDder perturbation−thSeory yields
R
PS= β/2dτ ϕ∗(τ)(∂τ+m−λ)ϕ(τ)+ λ2|ϕ(τ)|4 . (13) pˆ2 = 1cothβM βλcothβ2M , (15)
Z−β/2 h i h i 2 2 − 4sinh2 β2M
Inthissimplecase,theonlydifferencetothenaiveuseof
where M = m corresponds to the naive coherent state
coherentstatesistheshiftofλinthemassterm(besides
approachand M =m λ is our suggestion.
constant terms). −
IntableI,weshowhowthiscomparesforbothchoices
Only the symmetric ordering of (5) avoids unpleasant
of M and different values of m and β with the exact re-
1/N–corrections. This criterion leads to a unique trans-
sult. We take λ to be small against T = 1/β and m
lation into a continuous functional integral. We can also
in order to justify the perturbative approach. The re-
takethe perspectivefromthe problemofoperatororder-
sults clearly demonstrate thatthe shift inthe mass term
ing[14–17]. Tothinkoftheeasiestexample,thedifferent
is necessary for quantitatively correct results, and this
operatorsxˆpˆandpˆxˆgivethesamezerothordercontribu-
alone should be convincing that the correction we intro-
tionintheaction. Thisleadstoanambiguousfunctional
ducedtothestandardfunctionalintegralisnecessaryfor
integralif higher ordercorrectionsare not taken into ac-
thecorrectinterpretationoftheparametersintheaction.
count. The difference between these operators must oc-
cur elsewhere in the functional integral, and we see that TABLE I. Momentum squared hpˆ2i for two different sets
it precisely appears in the 1/N–terms. These can give of β and m by exact quantum mechanics, eq. (14), and by
finite contributionsif the evaluationofthe functionalin- first order perturbation theory from the functional integral,
tegral preceeds the limit N . In the transition to eq. (15). We compare the suggested shift in the mass term
→ ∞
the continuous functional integral, they would thus be (M=m−λ)withthenaivecoherentstateapproach(M=m).
erroneously neglected. Allentriesmustbedividedby103,andthezero–temperature
valueof 0.5 is subtracted from hpˆ2i.
hpˆ2i for β=1,m=5 hpˆ2i for β =3,m=1
λ
III. A SIMPLE TEST exact M=m−λ M=m exact M=m−λ M=m
0 6.7837 6.7837 6.7837 52.396 52.396 52.396
2 6.7835 6.7835 6.7698 52.361 52.359 52.030
Inordertocheckourassertionthatthequadraticterm
4 6.7833 6.7832 6.7560 52.327 52.320 51.665
should be m λ instead of m, we calculate the ther-
− 6 6.7831 6.7830 6.7421 52.293 52.278 51.299
mal equilibrium value of the momentum squared from
8 6.7829 6.7827 6.7283 52.259 52.232 50.934
the functional integralin firstorderperturbationtheory.
This is compared to a direct evaluation from ordinary
3
IV. FUNCTIONAL INTEGRAL FOR calculation of the critical chemical potential for the λ–
INTERACTING BOSONS transition of helium–4 confirm the necessity of this shift
and strengthen our faith in the suggested prescription.
The preceeding derivation for quantum mechanics can
Thesefindingsshouldbeimportanttorenormalization
directly be generalized to an interacting ensemble of
group treatments of statistical systems such as Bose–
bosons and therefore becomes relevant to the treatment
Einstein–condensation for interacting systems as well as
of the superfluid transition of helium–4 or Bose con-
to the understanding of functional integrals in general.
densation. We describe the system through the Hamil-
tonian operator (1), assuming the existence of a UV– We are grateful to Andrew Waldron for drawing our
momentum–cutoff Λ, q2 Λ2. The generalizationof our attentiontotheworksofChristandLeeandonthetreat-
proceduregivesthe actio≤n(3)with (apartfromconstant ment of the non–linear sigma model.
terms) v˜(q)=v(q), µ˜ =µ+Ω, and
1
Ω= v(0)+v(q) . (16)
2
q2X<Λ2(cid:18) (cid:19)
We obtaina cutoff–dependent “counterterm”Ω to the
∗ e–mail: [email protected]
chemical potential, which will be of importance for com-
† e–mail: [email protected]
parison with experiment. As it turns out, this cancels
[1] R. P. Feynman,Rev.Mod. Phys. 20, 367 (1948).
a similar term generated by the one loop approximation
[2] C. Garrod, Rev.Mod. Phys. 38, 483 (1966).
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We have calculated the chemical potential at the λ– (1973).
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elsewhere[19]. Herewejustrefertothevalueofthecriti-
[8] J.deBoer,B.Peeters,K.Skenderis,andP.vanNieuwen-
calchemicalpotential,whichcomesouttobe 6.75Kin
− huizen,Nucl.Phys.B446,211(1995);Nucl.Phys.B459,
good agreement with the experimental value of around
631 (1996); hep-th/9511141.
7.4 K [20]. As Ω in our case has a value of approxi-
− [9] M. Bijlsma and H. T. C. Stoof, Phys. Rev. A54, 5085
mately 12 K (with a cutoff given by the atomic length
(1996).
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V. CONCLUSION tegration (John Wiley, NewYork,1981).
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In this letter, we advocate a specific approach to
(1970).
the derivation of the functional integral for an interact-
[15] L. Cohen, J. Math. Phys. 11, 3296 (1970).
ing quantum model and its generalization to interacting
[16] M. M. Mizrahi, J. Math. Phys. 16, 2201 (1975).
bosonicsystems. Weareledtoashiftinthemasstermof
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theactionascomparedtoconventionalapproachesbased
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4