Monte Carlo evaluation of the continuum limit of ( ϕ 12 ) 3
aa r X i v : . [ h e p - l a t ] D ec Monte Carlo evaluation of the continuum limit of ( φ ) Riccardo Fantoni ∗ Universit`a di Trieste, Dipartimento di Fisica,strada Costiera 11, 34151 Grignano (Trieste), Italy (Dated: December 9, 2020)
Abstract
We study canonical and affine versions of non-renormalizable euclidean classical scalar field-theory with twelfth-order power-law interactions on three dimensional lattices through the MonteCarlo method. We show that while the canonical version of the model turns out to approach a“free-theory” in the continuum limit, the affine version is perfectly well defined as an interactionmodel.
PACS numbers: 03.50.-z,11.10.-z,11.10.Gh,11.10.Kk,02.70.Ss,02.70.Uu,05.10.LnKeywords: Monte Carlo method, euclidean scalar field-theory, power-law interactions, non-renormalizabletheory, free-theory, canonical version, affine version ∗ [email protected] . INTRODUCTION Classical versions of all covariant scalar field-theory models with positive interactionsadmit acceptable solutions, but some models will lead to divergences when trying to solvethem when using canonical quantum versions [1].Although classical covariant models, such as ( φ ) , lead to acceptable solutions, canonicalquantization leads only to free solutions, as if the interaction term was not present. There aresimple classical models, e.g., a half-harmonic oscillator which is limited to 0 < q < ∞ , thatalso fails using canonical quantization. A newer procedure, called affine quantization [2–4],differs from canonical quantization only because it promotes different canonical variables toquantum operators. It has been shown that affine quantization can successfully quantize theoscillator example, and the purpose of this paper is to demonstrate that affine quantization,in effect, just adds one additional term, which is proportional to ~ , to the Hamiltonian.Which extra term to add is guided by affine quantization, and the result leads to a validquantization of ( φ ) . The problem treated in this work deals with covariant scalar fields with power-law in-teractions. For the ( φ r ) d theory, the euclidean time version of the action functional is thengiven by, S [ φ ] = Z ( " s X µ =0 (cid:18) ∂φ ( x ) ∂x µ (cid:19) + m φ ( x ) + gφ r ( x ) ) d d x, (1.1)with x = ( x , x , . . . , x s ) for s spatial dimensions and d = s + 1 for the number of space-time dimensions, m is the bare mass, g is the interaction term coupling constant and r =4 , , , .... is the power of the interaction term.Monte Carlo (MC) [5–7] studies in 1982 [8] showed that these models were correct for r = 4 and d = 3 but when r = 4 and d = 4 they led only to free models, with a vanishingrenormalized coupling constant in the continuum limit, and this was confirmed later byanalytic studies and that even became simply free models when r = 4 and d > r ≥ d/ ( d − affine quantization procedures will solve those problems. The example ( φ ) has been deliberately chosen to be highly nonrenormalizable, while requiring the leastamount of computer time.
2n this work we chose the ( φ ) theory. Classically this is a straightfoward problemthat in the g → D g =0 , is larger than that ofthe interacting model D g> (integrating φ will be finite for less φ than in the free model).In the continuum limit the domains disagree and by continuity the new domain for the “free”version (we can call it a “pseudofree” situation) is the domain D g> , not D g =0 . That is thesource of having free models using canonical quantization such as ( φ r ) d with r > d/ ( d − . On the other hand, affine quantization will lead to a non-free model to begin with andso it is appropriate when g →
0. In parallel to the covariant theory one can define also an ultralocal theory which is obtained by neglecting the kinetic part of the action (the term P µ ( ∂φ ( x ) /∂x µ ) ) [3]. It turns out that such a theory will have a divergent perturbationseries already for r > d ≥
2. In these cases the field theory will lead to a free-theory,non-renormalizable. So, with r = 12 there should be an even greater difference between thecanonical and affine versions. II. AFFINE VERSION OF THE FIELD-THEORY
Our model has a standard classical Hamiltonian given by, H [ π, φ ] = Z ( " π ( x ) + s X µ =1 (cid:18) ∂φ ( x ) ∂x µ (cid:19) + m φ ( x ) + gφ r ( x ) ) d s x, (2.1)where s denotes the number of spatial coordinates and x is the time. The momentum field π ( x ) = ∂φ ( x ) /∂x and the canonical action S = R H dx is the one of Eq. (1.1).Next, we introduce the affine field κ ( x ) ≡ π ( x ) φ ( x ), with φ ( x ) = 0 and modify theclassical Hamiltonian to become [2–4], H ′ [ κ, φ ] = Z ( " κ ( x ) φ − ( x ) κ ( x ) + s X µ =1 (cid:18) ∂φ ( x ) ∂x µ (cid:19) + m φ ( x ) + gφ r ( x ) ) d s x. (2.2)In an affine quantization the operator term b κ ( x ) φ − ( x ) b κ ( x ) = b π ( x ) + ~ (3 / δ s (0) φ − ( x )which leads to an extra “3 /
4” potential term (see Appendix A). So that the new affine actionwill read, S ′ [ φ ] = Z ( " s X µ =0 (cid:18) ∂φ ( x ) ∂x µ (cid:19) + m φ ( x ) + gφ r ( x ) + 38 ~ δ s (0) / [ φ ( x ) + ǫ ] ) d d x, (2.3) One requires that R dx [ ∇ φ ( x )] < [ R dx φ r ( x )] /r . ǫ > /
4” extra term. In the g → / ~ δ s (0) / [ φ ( x ) + ǫ ]interaction term. III. THE LATTICE FORMULATION OF THE FIELD-THEORY MODEL
We used a lattice formulation of the field theory. The theory considers a real scalarfield φ taking the value φ ( x ) on each site of a periodic, hypercubic, d -dimensional lattice oflattice spacing a and periodicity na . The canonical action for the field, Eq. (1.1), is thenapproximated by S [ φ ] ≈ ( "X x,µ a − ( φ ( x ) − φ ( x + e µ )) + m X x φ ( x ) + g X x φ r ( x ) ) a d , (3.1)where e µ is a vector of length a in the + µ direction. The vacuum expectation of a functionalobservable F [ φ ] is h F i ≈ R F [ φ ] exp( − S [ φ ]) Q x dφ ( x ) R exp( − S [ φ ]) Q x dφ ( x ) . (3.2)We will approach the continuum limit by choosing na = 1 fixed and increasing thenumber of discretizations n of each component of the space-time. So that the lattice spacing a = 1 /n → . IV. SIMULATION DETAILS AND RELEVANT OBSERVABLES
From each real field φ ( x ) we extract the Fourier transform e φ ( p ) = Z d d x e ip · x φ ( x ) , (4.1)with e φ ∗ ( p ) = e φ ( − p ), so that the action of Eq. (1.1) becomes S [ e φ ] = Z
12 [ p + m ] | e φ ( p ) | d d p (2 π ) d + gI r [ e φ ] , (4.2)where we denote with I r the power-law interaction functional. Note that one could change the field φ → φ ′ a − d/ so that for example the kinetic term of the action goesto simply P x,µ [ φ ′ ( x ) − φ ′ ( x + e µ )] /
4e then find the ensemble averages h e φ (0) i and h e φ (0) i and construct the followingobservable (a renormalized unitless coupling constant at zero momentum), g R = 3 h e φ (0) i − h e φ (0) ih e φ (0) i , (4.3)so that clearly, using path integrals in the Fourier transform of the field, we immediatelyfind for the canonical version of the theory, g R g → −→ . (4.4)This remains true even for the calculation on a discrete lattice.We then choose the momentum p with one component equal to 2 π/na and all othercomponents zero and calculate the ensemble average h| e φ ( p ) | i . We then construct the renor-malized mass m R = p h| e φ ( p ) | ih e φ (0) i − h| e φ ( p ) | i . (4.5)When g = 0 the canonical version of the theory can be solved exactly yielding m R g → −→ [ π/n sin( π/n )] m. (4.6)Following Freedman et al. [8] we will call g R a dimensionless renormalized couplingconstant and we will use it to test the “freedomness” of our field theories in the continuumlimit. Note that the sum-rules of Eqs. (4.4) and (4.6) do not hold for the affine version (2.3)of the field theory due to the additional (3 / ~ δ s (0) / [ φ ( x ) + ǫ ] interaction term.Our MC simulations use the Metropolis algorithm [5, 7] to calculate the discretize versionof Eq. (3.2) which is a n d multidimensional integral. The simulation is started from theinitial condition φ = 0. One MC step consisted in a random displacement of each one of the n d components of φ as follows φ → φ + ( η − / δ, (4.7)where η is a uniform pseudo random number in [0 ,
1] and δ is the amplitude of the displace-ment. each one of these n d moves is accepted if exp( − ∆ S ) > η where ∆ S is the change inthe action due to the move (it can be efficiently calculated considering how the kinetic partand the potential part change by the displacement of a single component of φ ) and rejectedotherwise. The amplitude δ is chosen in such a way to have acceptance ratios as close as5ossible to 1 / N = 10 steps. The statistical error on the average h F i will then depend onthe correlation time necessary to decorrelate the property F , τ F , and will be determined as p τ F σ F / ( N n d ), where σ F is the intrinsic variance for F . V. SIMULATION RESULTS
We first chose the euclidean covariant scalar interaction model with d = 3 and r = 12.In its canonical version (see the action of Eq. (1.1)) this is a non-renormalizable model and,following a perturbation expansion of g there is an infinite number of different, divergentterms; or, if treated as a whole, such model collapses to a “free-theory” with a vanishinginteraction term [9, 10]. This is even more true for the ultralocal version of the theory.Following Freedman et al. [8], in our MC simulation, for each n and g , we adjustedthe bare mass m in such a way to maintain the renormalized mass approximately constant m R ≈ g was necessary to take a complex bare mass so that m was negative), towithin a few percent (in all cases less than 15%), and we measured the renormalized couplingconstant g R of Eq. (4.3) for various values of the bare coupling constant g at a given smallvalue of the lattice spacing a = 1 /n . Thus with na and m R fixed, as a was made smallerwhatever change we found in g R m dR as a function of g could only be due to the change in a . We generally found that a depression in m R produced an elevation in the correspondingvalue of g R and viceversa. The results are shown in Fig. 1 for the covariant version, where,following Freedman et al. [8] we decided to compress the range of g for display by choosingthe horizontal axis to be g/ (50 + g ). As we can see from the figure the renormalized masswas made to stay around a value of 3, even if this constraint was not easy to implementsince for each n and g we had to run the simulation several (5-10) times with different valuesof the bare mass m .In Fig. 2 we show the same calculation but for the regularized affine field-theory (see theaction of Eq. (2.3)) where we take ~ = 1 and ǫ = 10 − .From Fig. 1 we can see how at all finite value for the bare coupling constants g therenormalized coupling at zero momentum g R m dR appears to move to zero uniformly as thelattice spacing gets small, for n → ∞ . This numerically proves that the canonical theory be-comes asymptotically a free-theory in the continuum limit of large n . Which is in agreement6 .52.62.72.82.93.03.13.23.33.40.0 0.2 0.4 0.6 0.8 1.0 m R g/(50+g)covariant canonical d=3, r=12 n=4n=6n=10n=12 −50510152025300.0 0.2 0.4 0.6 0.8 1.0 g R ( m R ) d g/(50+g)covariant canonical d=3, r=12 n=4n=6n=10n=12 FIG. 1. (color online) We show the renormalized mass m R of Eq. (4.5) (top panel) and therenormalized coupling constant g R m dR of Eq. (4.3) (bottom panel) as calculated from Eq. (3.2)for m R ≈ g at decreasing values of the latticespacing a = 1 /n ( n → ∞ continuum limit) for the canonical ( φ ) euclidean scalar field theorydescribed by the action in Eq. (1.1). The lines connecting the simulation points are just a guidefor the eye. with the well known theoretical results [2–4]. This does not happen for the affine theory asshown in Fig. 2, where the renormalized coupling of the theory stays far from zero in thecontinuum limit for all values of the bare coupling constant.7 .42.62.83.03.23.43.63.80.0 0.2 0.4 0.6 0.8 1.0 m R g/(50+g)covariant affine d=3, r=12 n=4n=6n=10n=12 g R ( m R ) d g/(50+g)covariant affine d=3, r=12 n=4n=6n=10n=12 FIG. 2. (color online) We show the renormalized mass m R of Eq. (4.5) (top panel) and therenormalized coupling constant g R m dR of Eq. (4.3) (bottom panel) as calculated from Eq. (3.2) for m R ≈ g at decreasing values of the lattice spacing a = 1 /n ( n → ∞ continuum limit) for the affine ( φ ) euclidean scalar field theory described bythe action in Eq. (1.1). The lines connecting the simulation points are just a guide for the eye. VI. CONCLUSIONS
Using MC simulations, we determined the dimensionless renormalized coupling constantof an euclidean classical scalar field-theory with twelfth-order power-law interactions ona three dimensional lattice. It turns out that the canonical version of the theory has anoninteracting continuum limit. The renormalized coupling constant tends to zero at each8nite value of the bare coupling constant as the lattice spacing gets small.We then formulated an affine version of the same field-theory with the “3 /
4” interactionterm and observed that the MC results for the renormalized coupling constant stays farfrom zero for all values of the bare coupling constant as the lattice spacing diminishes. Thismeans that the affine model remains a well defined interacting model in the continuum limit.A classical model such as ( φ ) with a positive coupling constant has a natural behavior,while it becomes a free theory with a positive coupling constant using canonical quantization.Canonical quantization also fails for a half-harmonic oscillator, e.g., 0 < q < ∞ as well.Affine quantization solves both of these problems. There is a genuinely new procedure thatpermits various problem models to achieve a proper quantization. Affine quantization justselects different classical variables to promote to operators, and then it proceeds just likecanonical quantization thereafter.The present paper shows that the model ( φ ) also generates a nontrivial behavior withan affine quantization. It is designed to feature a region where canonical quantization failsand there is a new procedure that can help. The classical limit of this quantized modelleads back to a classical model with a positive coupling constant. That does not happenfor canonical quantization. This implies that while canonical quantization is good for somemodels, affine quantization is needed for other models.There are many other models that canonical quantization cannot solve, or struggle toquantize, that may be possible to quantize using affine quantization. Some of those mod-els may be useful to specific problems in present-day high energy physics. This paper ispurposely designed to open a new door to high energy physics and in the field of quantumgravity [2, 3]. Appendix A: The origin of the “ / ” extra term The operator corresponding to the affine field κ will be the dilation operator b κ = ( b π b φ + b φ b π ) / b φ = φ and b π = − i ~ δ φ = − i ~ δ/δφ so that the commutator [ b φ ( x ) , b π ( y )] = i ~ δ s ( x − y ), where δ s ( x ) is a s -dimensional Dirac delta function since δ φ ( x ) φ ( y ) = δ s ( x − y ). Multiplying thisby b φ we find [ b φ, b φ b π ] = [ b φ, b π b φ ] = [ b φ, b κ ] = i ~ δ s b φ which is only valid for φ = 0. Then b κ = − i ~ { δ φ ( x ) [ φ ( x )] + φ ( x ) δ φ ( x ) } / − i ~ { δ s (0) / φ ( x ) δ φ ( x ) } . Now, for φ ( x ) = 0, we will9ave that affine quantization sends b π ( x ) to b κ ( x ) φ − ( x ) b κ ( x ) = − ~ { δ s (0) / φ ( x ) δ φ ( x ) } φ − ( x ) { δ s (0) / φ ( x ) δ φ ( x ) } = − ~ { δ s (0) φ − ( x ) / δ s (0) φ ( x ) δ φ ( x ) [ φ − ( x )] / δ s (0) φ − ( x ) δ φ ( x ) / δ s (0) φ − ( x ) δ φ ( x ) / − δ s (0) φ − ( x ) δ φ ( x ) + δ φ ( x ) } = − ~ { δ s (0) φ − ( x ) / − δ s (0) φ − ( x ) / δ φ ( x ) } = ~ (3 / δ s (0) φ − ( x ) − ~ δ φ ( x ) = ~ (3 / δ s (0) φ − ( x ) + b π ( x ) . (A1)We then see the appearance of an extra “3 /
4” potential term. The lattice version of suchterm will then be ~ (3 / a − s φ − ( x ) (A2)where a is the lattice spacing. ACKNOWLEDGMENTS
I would like to thank John R. Klauder who proposed the problem, for the lively discussionexchanges. I would also like to acknowledge the constant support of my wife Laure Goubawho introduced me to the method of affine quantization. [1] John. R. Klauder,
A Modern Approach to Functional Integration (Springer, 2010).[2] J. R. Klauder, “Using affine quantization to analyze non-renormalizable scalar fields and thequantization of einstein’s gravity,” (2020), arXiv:2006.09156.[3] J. R. Klauder, “An ultralocal classical and quantum gravity theory,” Journal of High EnergyPhysics, Gravitation and Cosmology , 656 (2020).[4] J. R. Klauder, “The benefits of affine quantization,” Journal of High Energy Physics, Gravi-tation and Cosmology , 175 (2020).[5] M. H. Kalos and P. A. Whitlock, Monte Carlo Methods (Wiley-Vch Verlag GmbH & Co.,Germany, 2008).[6] M. P. Allen and D. J. Tildesley,
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7] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. M. Teller, and E. Teller, “Equationof state calculations by fast computing machines,” J. Chem. Phys. , 21 (1953).[8] B. Freedman, P. Smolensky, and D. Weingarten, “Monte carlo evaluation of the continuumlimit of φ and φ ,” Physics Letters , 481 (1982).[9] M. Aizenman, “Proof of the triviality of φ d field theory and some mean-field features of isingmodels for d > , 886(E) (1981).[10] J. Fr¨ohlich, “On the triviality of λφ d theories and the approach to the critical point in d ≥ , 281 (1982). .40.50.60.70.80.91.01.10.0 0.2 0.4 0.6 0.8 1.0 g R g/(50+g)covariant affine d=3, r=12 n=4n=6n=10n=12 g R g/(50+g)covariant canonical d=3, r=12g/(50+g)covariant canonical d=3, r=12