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\centerline{\bf Poisson Distributions for $\bf\varphi^4$}
\centerline{by}
\centerline{John R. Klauder}
\centerline{\it Department of Physics}
\centerline{\it University of Canterbury}
\centerline{\it Christchurch, New Zealand}
\centerline{and}
\centerline{\it Departments of Physics and Mathematics}
\centerline{\it University of Florida}
\centerline{\it Gainesville, FL 32611$^*$}
\vskip .5truein
\centerline{\bf ABSTRACT}
\doublespace
In an attempt to avoid triviality for relativistic quantum $\varphi^4_n$
theories for space-time dimensions $n\geq 5$, and possibly $n=4$ as
well, an additional,
nonclassical, nonpolynomial, local potential is included, along with
standard factors, in a lattice-regularized formulation of the model. It
is argued that if the additional term redistributes the field
probability in the manner characteristic of a generalized Poisson
distribution, then a nontrivial quantization may be achieved, one which
also passes to the correct nontrivial classical theory in the
appropriate limit.
\vfill
$^*$Permanent Address
\eject
All scalar fields $\phi^4_n$ in space-time dimensions $n\geq 5$ (and
probably $n=4$ as well) have the feature of being classically nontrivial
[1] while their present quantum formulations are trivial (Gaussian) [2].
Such trivial quantum results may be understood as the consequence of forcing
nonasymptotically-free field theories into the straightjacket of
standard lattice formulations that are inherently consistent with
asymptotic freedom. Alternatively stated, trivial quantum results arise
for such theories in the continuum limit because in high enough
space-time dimensions the non-Gaussian parts of the correlations between
fields on neighboring lattice sites are simply too weak to withstand the
stronger Gaussian tendencies implicit in the central limit theorem. In
this letter we propose an alternative lattice-space formulation for such
models---and implicitly for a wide variety of other models as
well---that promises to yield nontrivial quantum results, and which
should also have the known, nontrivial classical behavior in the
appropriate classical limit. In fact, our proposal may well offer
alternative, nontrivial quantizations for conventional nontrivial
models, such as $\phi^4_n$, $n\leq 3$; however, it is for $n\geq 5$
($n=4$) that our method should lead to nontrivial quantizations rather
than the physically unacceptable trivial results presently available.
In the course of the next few paragraphs, we introduce in several stages
the various features that characterize our models.
We begin by showing the equivalence of two distinct versions of the
classical theory. Conventionally, one begins with a classical field
$\varphi_{c\ell}(x)$, $x\in \ir^n$, and an action functional
$$\int \left(h(x) \varphi_{c\ell}(x)+{1 \over 2}\left\{[\partial_\mu
\varphi_{c\ell}(x)]^2-m^2\varphi_{c\ell}^2(x)\right\}-
g\varphi_{c\ell}^4(x)\right)d^nx.$$
The associated equation of motion
$$(\square +m^2)\varphi_{c\ell}(x)+4g\varphi_{c\ell}^3(x)=h(x)$$
admits a solution we denote by
$\varphi_{c\ell}(x;\varphi_{in},\dot\varphi_{in},h)$ that depends on the
initial data
$\varphi_{in}(\hbox{\bf x}),\dot\varphi_{in}(\hbox{\bf x})$
(say at time $t=0$), and the
source $h$. As a second choice we begin with a field $\Phi_{c\ell}(x,w)$
which also depends on an auxiliary, dimensionless variable $w \in
(0,1)$, and for this field we adopt the action functional
$$\int \left(h(x)\Phi_{c\ell}(x,w)+{1 \over 2}\left\{[\partial_\mu
\Phi_{c\ell}(x,w)]^2-m^2\Phi_{c\ell}^2(x,w)\right\}-
g\Phi_{c\ell}^4(x,w)\right)d^nx dw;$$
note that the derivatives are still only with respect to $x$ and none appear
for $w$. The associated equation of motion
$$(\square +m^2)\Phi_{c\ell}(x,w)+4g\Phi_{c\ell}^3(x,w)=h(x) ,$$
coupled with the particular initial data $\Phi_{in}(\hbox{\bf
x},w)=\varphi_{in}(\hbox{\bf x})$
and $\dot\Phi_{in}(\hbox{\bf x},w)=\dot\varphi_{in}(\hbox{\bf x})$ lead
to a solution
$$\Phi_{c\ell}(x,w;\Phi_{in}\dot
\Phi_{in},h)=\varphi_{c\ell}(x;\varphi_{in},\dot\varphi_{in},h) ,$$
with all relations holding for $w \in (0,1)$. In brief, if one cannot
``initiate'' or ``test for'' any $w$ dependence, then the two action
functionals lead to equations of motion with identical classical
solutions. The present letter examines the quantum theory of the second
(``diastrophic'' [3]) theory as a quantum model for $\varphi^4_n$.
A formal expression for a path integral quantization may be readily
obtained. If
$$\eqalign{Z\{h\}&\equiv \langle 0| Te^{i\int h(x) \varphi(x) d^nx}
|0\rangle\cr
&= \langle 0| Te^{i\int h(x)\Phi(x,w) d^nx\,dw}|0\rangle\cr}$$
denotes the generating functional of time-ordered Green functions for
the field $\varphi(x) \equiv \int \Phi(x,w)dw$, then $Z$ admits a formal
functional integral representation given by
$$\eqalignno{Z\{h\} &= {\cal N} \int\exp\Biggl (i\int h(x)\Phi (x,w)
d^nxdw + i\int \biggl \{ {\scriptstyle 1\over 2}
\left[ \left(\partial_\mu\Phi (x,w)\right )^2 -m^2\Phi^2(x,w)\right
]\biggr.\Biggr.\cr
&\Biggl.\biggl.\quad - g\Phi^4(x,w)-P[\Phi(x,w)]\biggr \}
d^nxdw\Biggr)\,
\Pi_{x,w} d\Phi(x,w)\, ;\cr}$$
the additional (nonpolynomial) potential $P[\Phi]$ $(=P[-\Phi])$ is
discussed below. A corresponding expression for the Euclidean-space
generating functional is formally given by
$$\eqalign{S\{h\}&={\cal N} \int \exp \Biggl(\int h(x)\Phi (x,w) d^nxdw
-\int
\biggl\{ {\scriptstyle 1\over 2} [(\nabla\Phi(x,w))^2
+m^2\Phi^2(x,w)]\biggr.
\Biggr.\cr
&\Biggl.\biggl.\quad + g\Phi^4(x,w)+P[\Phi(x,w)]\biggr\} d^nxdw\Biggr )
\, \Pi_{x,w}d\Phi(x,w)\, ,\cr}$$
which, in turn, may be given an $x$ {\it and} $w$, $n$-(hyper)cubic
$\times$ linear lattice-space formulation as the continuum (and
infinite space-time volume) limit implicit in the expression
$$\eqalignno{S\{h\}&= \lim_{\epsilon,a\to 0} \prod_{r=1}^R N \int
\exp [\Sigma h_k \Phi_{kr} a^n \epsilon -{\scriptstyle 1\over 2}
Y (a,\epsilon) \Sigma (\Phi_{k^*r} -\Phi_{kr})^2 a^{n-2}\epsilon \cr
&\quad - {\scriptstyle 1\over 2} m_0^2 (a,\epsilon)
\Sigma\Phi^2_{kr} a^n\epsilon - g_0 (a,\epsilon) \Sigma\Phi^4_{kr}
a^n\epsilon - \Sigma P (\Phi_{kr}, a,\epsilon) a^n\epsilon ]
\Pi_k d\Phi_{kr}\, .\cr}$$
Here $\epsilon$ $(\equiv R^{-1})$ is the lattice spacing in $w$, $r\in
\{1,2,\ldots\}$ labels a lattice site in $w$ space, $a$ is the lattice
spacing in $x$, $k=(k_1,\ldots,k_n), k_j\in \{0,\pm 1,\ldots\},$ labels
a lattice site in $x$ space, and $k^*$, and an implicit sum, includes
$1\over 2$ the nearest neighbors of $k$, as usual. We have also
anticipated and introduced cutoff-dependent coefficients $Y,m_0,g_0$, as
well as for the as-yet-unspecified term $P$.
Clearly, any $r$-dependence of the integration variables $\Phi_{kr}$
is irrelevant, and in fact the expression prior to taking the limit is
really the $R$th power of a ``base-theory'' integral. Observe that $R$
then enters in a manner similar to the number of ``replicas'' in statistical
physics. Although that number originated from the limit of integration
for $w$, we shall find it expedient to relax that condition (for $n\ge
4$) and allow for replica number renormalization by hereafter replacing
that number by $\bar{R}\equiv [\bar\epsilon^{ -1}]\equiv
[\epsilon^{-1}Q(a)^{-1}]$ for some ($n$-dependent) $Q(a)\le 1$, where
[$A$] denotes the integer part of $A$. As a consequence
$$\eqalign{S\{h\} &=
\lim_{\epsilon,a\to 0}
[s(h)]^{\bar{R}}\, ,\cr
s(h)&= N\int \exp
[ \Sigma h_k \Phi_k
a^n\epsilon
- {\scriptstyle 1\over 2}
Y(a,\epsilon)
\Sigma (\Phi_{k^*} -\Phi_k)^2
a^{n-2}
\epsilon\cr
&\quad
-{\scriptstyle 1\over 2}
m_0^2
(a,\epsilon)
\Sigma
\Phi_k^2
a^n\epsilon
-g_0
(a,\epsilon)
\Sigma
\Phi_k^4
a^n\epsilon
-\Sigma P
(\Phi_k, a, \epsilon)
a^n\epsilon ] \Pi d\Phi_k\, .\cr}$$
Apart from the term $P$ and the appearance of the parameter $\epsilon$,
the expression for $s(h)$ is similar to a conventional lattice-space
formulation for the model in question [4]. We can make that similarity
even closer if we next assume that $Y(a,\epsilon)=Y (a)\epsilon$,
$m_0^2(a,\epsilon) = m_0^2(a)\epsilon$, and $g_0(a,\epsilon) =
g_0(a)\epsilon^3$; this dependence on $\epsilon$ reflects the
multiplicative renormalization found necessary in previous operator
treatments of such models [3]. A simple variable change then leads to
our final integral representation for $s(h)$ given by
$$\eqalign{s(h)&=
N_0\int \exp [ \Sigma h_k \phi_k a^n-{\scriptstyle 1\over 2} Y(a) \Sigma
(\phi_{k^*} - \phi_k)^2 a^{n-2}\cr &\quad -{\scriptstyle 1\over 2}
m_0^2(a)\Sigma \phi_k^2 a^n -g_0 (a) \Sigma \phi_k^4 a^n -\Sigma P_0
(\phi_k,a,\epsilon)a^n] \Pi d\phi_k\, ,\cr}$$
where we have introduced
$P_0$ which now carries the only dependence on $\epsilon$ within $s(h)$.
If it were not for $P_0$ this expression would exactly resemble the
standard lattice formulation.
The factor $N_0$ is chosen so that $s(0)=1$, and as a consequence the
expression for $s(h)$ assumes the alternative form
$$s(h)=\exp \Biggl [\sum_{\ell=1}^\infty (\ell !)^{-1}
\sum h_{k_1}\cdots h_{k_\ell} \langle \phi_{k_1}\cdots
\phi_{k_\ell}\rangle^T a^{\ell n}\Biggr ]\, ,$$
where averages $\langle \cdot \rangle$ are defined in the
$\epsilon$-dependent probability distribution implicit in the expression
$s(h)\equiv \langle \exp (\Sigma h_k \phi_k a^n)\rangle$. Since the
ultimate answer of interest involves $[s(h)]^{\bar{R}}$, and
$\bar{R}\to \infty$ as $\bar{\epsilon}\to 0$, a nontrivial
result emerges, when $\bar{\epsilon} \ll 1$, provided the truncated
correlation functions satisfy
$$\langle \phi_{k_1}\phi_{k_2}\cdots \phi_{k_\ell}\rangle^T\propto
\bar{\epsilon}$$
for all $\ell \geq 1$; with $P_0$ and the rest of the potential even,
only the even-order (truncated) moments are nonzero in the present case.
The advantage of arranging this proportionality to $\bar{\epsilon}$
will become clear shortly.
To this end, let us first recall the usual definition [4,5] of the
dimensionless renormalized coupling constant $g_R\equiv$ lim $g(a)$ for
the base theory given as a space-time continuum and infinite volume
limit of the expression
$$g(a)=-{\Sigma_{k\ell m} \langle \phi_0 \phi_k\phi_\ell \phi_m\rangle^T
\over [ \Sigma_k\langle \phi_0\phi_k\rangle ]^2 [\Sigma_\ell
\ell^2\langle\phi_0\phi_\ell \rangle /\Sigma_m\langle
\phi_0\phi_m\rangle ]^{n/2}}\, .$$
Let us initially ignore the amplitude dependence and present the usual
\hbox{argument} for triviality. From the viewpoint of critical
phenomena [5], it follows that
$g(a)\propto a^{(\gamma +n\nu -2\Delta)/\nu}$ for small $a$. For $n\geq
5$, when the critical exponents follow from mean field theory,
$g(a)\propto a^{(n-4)}$, an expression which tends to zero as $a\to 0$
leading to triviality. For $n=4$, renormalization group arguments assert
that $g(a)\propto 1/|$ln$(\mu a)|$, for some mass parameter $\mu$,
which as $a\to 0$ also leads to
triviality. For $n\leq 3$ hyperscaling ensures that $g(a) = O(1)$ and
thus $g_R$ is nonvanishing.
Next let us pursue the consequences of including the proper amplitude
factors of the truncated correlation functions. For $n\geq 5$,
$g(a)\propto \epsilon^{-1} Q(a)^{-1} a^{(n-4)}$, and so for $n\geq 5$
we choose $Q(a)=[\mu a/(1+\mu a)]^{(n-4)}$ with $\mu$ a suitable mass
parameter. For $n=4$ we choose $Q(a)=1/(|$ln$(\mu a)|+1)$, while
for $n\leq 3$ we set $Q(a)\equiv 1$. Therefore, for every $n\geq 1$, it
follows that $g(a)\propto \epsilon^{-1}$ and consequently, as $a\to 0$,
$g_R\propto \epsilon^{-1}$ and is nonvanishing as well.
The purpose of the term $P_0$ is to ensure the required dependence of
the truncated correlation functions on $\bar\epsilon$. When
$\bar\epsilon$ is small it follows that
$$\langle\phi_{k_1}\cdots\phi_{k_\ell}\rangle =
\langle\phi_{k_1}\cdots\phi_{k_\ell}\rangle^T+O(\bar\epsilon^2)
= O(\bar\epsilon)\, ,$$
and thus the role of $P_0$, as well as the remaining model parameters
($Y$, $m_0$, and $g_0$) in the single-lattice-site distribution is,
roughly speaking, to divide the total probability distribution into the
sum of two contributions: (i)~one term, a highly concentrated
distribution, largely determined by $P_0$, where $\phi_k\simeq 0$ with a
total probability of $1-O(\bar\epsilon)$; and (ii)~the other term, a
nonconcentrated distribution, largely determined by the remaining model
parameters, where $\phi_k$ takes on general values but with a total
probability $O(\bar\epsilon)$. Such a division of probabilities is
exactly how various Poisson distributions avoid the Gaussian vise-grip
of the central limit theorem [6].
It is entirely reasonable that an extra term should appear in the
renormalized lattice action the purpose of which is to reduce the
probability of large field values. Recall, for classical functions,
that the Sobolev inequality
$$[ \int \phi^4 (x) d^nx ]^{1\over 2} \leq
C\int [ (\nabla \phi(x))^2 + m^2\phi^2 (x) ] d^nx$$
holds for finite $C$
whenever $n\leq 4$, but fails to hold for any $C<\infty$ whenever $n\geq
5$ [7]. Elsewhere we have interpreted such inequalities to imply that
any nonrenormalizable interaction (e.g., $\phi^4_n$, $n\geq 5$) acts
partially as a {\it hard core} in function space projecting out certain
field histories otherwise allowed if the interaction had been entirely
absent [8]. Ignoring fluctuations, the troublesome and so-projected
fields have local singularities [e.g., for $|x|<1$] of the form $\phi
\sim |x|^{-\gamma}$, where $(n/4) < \gamma < (n-2)/2$; these large
amplitude fields necessarily involve high momenta, but not all fields
involving high momenta [e.g., $\phi \sim \cos (x^{-2})$] are troublesome
in this sense. Thus a reasonable renormalization for such fields should
reweight their distribution in the manner indicated.
An indication of the general form of $P_0$ that should accomplish our
goal may be gleaned from the study of the explicitly soluble case for
$n=1$ (i.e., Euclidean quantum mechanics, and so we set $x=t$) [9]. In
this case $Q(a) \equiv 1$, but even after $a \to 0$, we are still able
to choose $P_0(\phi,\epsilon)$ so that $\langle \phi_{k_1}\cdots
\phi_{k_\ell}\rangle^T \propto \epsilon$. In particular, for $n=1$ and
any $\gamma$, ${1\over 2}< \gamma < {3\over 2}$, we can, after letting
$a\to 0$, choose [9]
$$\eqalign{P_0(\phi,\epsilon) &= {1\over 2}
{[\gamma (\gamma +1) \phi^2 -\gamma \delta^2 (\epsilon) ]\over [\phi^2
+\delta^2(\epsilon)]^2}\, ,\cr
\delta^2 (\epsilon)&= (G\epsilon)^{2/(2\gamma -1)}\, ,\cr
G&= \sqrt{\pi} \Gamma
(\gamma-{\scriptstyle 1\over 2} ) /\Gamma (\gamma)\, .\cr}$$
Observe in this case that $P_0$ is not unique. Nevertheless each such
$P_0$ may be interpreted as a regularized form of a formal interaction
proportional to $\int \Phi^{-2}(t,w) dw$, which in turn should be viewed
as a (decidedly unconventional) renormalization counterterm for $\int
\dot\Phi^2(t,w)dw$ rather than for the quartic interaction. This
interpretation is borne out by the fact that $P_0$ does not vanish when
$g_0\to 0$, meaning that the zero-coupling limit of the interacting
theory, which retains the effects of the hard core, is not the free
theory but a so-called pseudofree theory; it is expected that a
perturbation theory in the quartic interaction exists about the
pseudofree theory [8]. As a renormalization counterterm for the kinetic
energy, $P_0$ contains an implicit multiplicative factor of $\hbar^2$,
which in the classical limit, $\hbar\to 0$, implies that the expected
classical theory emerges as desired. For $n=1$ this expected behavior
has been confirmed [10].
For $n\geq 2$, and for a suitable ($n$-dependent) choice of $A$, $B$,
and $C$, it is suggestive that $P_0$ has the form
$$P_0 (\phi,a,\epsilon) ={A(a,\epsilon)\phi^2 -B(a,\epsilon)\over
[\phi^2 + C(a,\epsilon)]^2}$$
based on what holds for $n=1$. Although we have no real evidence for
this hypothesis, it is clear that an expression of this form will
accomplish the stated purposes, and it has the further advantage that it
too may be interpreted as a renormalization counterterm for the kinetic
energy. The coefficients must be chosen so that, like the case for
$n=1$, the distribution is a generalized Poisson process, rather than
just a compound Poisson process [6]. It may even be possible to
determine $P_0$ to a certain degree based on high-temperature series
expansions that exist for a general, even, single-site field
distribution [5]. One may try to determine the necessary
$\bar\epsilon$-dependence of the moments of the single-site field
distribution which ensures that the truncated correlation functions are
proportional to $\bar\epsilon$ when $\bar\epsilon\ll 1$; for this
purpose it is sufficient to focus on the second- and fourth-order
truncated correlation functions. In this way one may be able to suggest
a suitable $P_0(\phi,a,\epsilon)$ that leads to the required behavior of
the moments. Given a candidate choice for $P_0(\phi,a,\epsilon)$,
Monte-Carlo methods and/or renormalization-group techniques may then be
introduced. Although high-temperature series exist only for $n\leq 4$,
the principles set forth in this letter may be applied to study
$\phi_4^4$ and $\phi_4^6$, or even $\phi_3^4$, $\phi_3^6$, and
$\phi_3^8$, which include super renormalizable, renormalizable and
nonrenormalizable examples.
The resultant field will be infinitely divisible in the sense of
probability theory, as indeed are all Poisson distributions or limits
thereof [6]. A study of such fields has been made previously [11] where
it was shown that under certain assumptions such fields had vanishing
truncated four-point functions and an $S$-matrix of unity. Among those
assumptions was asymptotic completeness, a property that is missing for
the present models. Indeed, we have made the achievemant of a
nonvanishing four-point function our central goal, and granting
reasonable spectral properties of the theory, we expect to attain
nontrivial scattering. This would be a necessary precondition to have a
classical limit that agreed with the known proper nontrivial classical
theory.
\vskip .5truein
\centerline{\bf ACKNOWLEDGEMENTS}
The author thanks the Department of Physics of the University of
Canterbury for their hospitality and for the financial support of an
Erskine Fellowship.
\vskip .5truein
\centerline{\bf REFERENCES}
\item{1.} M. Reed, {\it Abstract Non Linear Wave Equations},
Lecture Notes in Mathematics Vol. 507 (Springer-Verlag, Berlin, 1976).
\item{2.} G. Gallovotti, Rev. Mod. Phys. {\bf 57}, 471 (1985);
D.J.E. Callaway, Phys. Rep. {\bf 167}, 241 (1988); V. Rivasseau,
{\it From Perturbative to Constructive Renormalization} (Princeton
Unversity Press, Princeton, 1991); R. Fern\'andez, J. Fr\"olich, and
A.D. Sokal, {\it Random Walks, Critical Phenomena, and Triviality in
Quantum Field Theory} (Springer-Verlag, New York, 1992).
\item{3.} J.R. Klauder, Phys. Rev. Lett. {\bf 28}, 769 (1972).
\item{4.} J. Glimm and A. Jaffe, {\it Quantum Physics}
(Springer-Verlag, New York, 1987).
\item{5.} G.A. Baker, Jr. and J.M. Kincaid, J. Stat. Phys.
{\bf 24}, 469 (1981).
\item{6.} B. de Finetti, {\it Theory of Probability},
Vol. 2 (John Wiley \& Sons, London, 1975).
\item{7.} O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva,
{\it Linear and Quasi-Linear Equations of Parabolic Type},
Trans. of Math. Mono., Vol. 23 (Amer. Math. Soc.,
Providence, Rhode Island, 1968).
\item{8.} J.R. Klauder, Acta Physica Austriaca, Suppl.
{\bf XI}, 341 (1973); Phys. Lett. {\bf 47B}, 523 (1973);
Ann. of Phys. {\bf 117}, 19 (1979).
\item{9.} J.R. Klauder, ``Self-Interacting Scalar Fields
and (Non-) Triviality,'' to appear in Proceedings of the Workshop
on Mathematical Physics Toward the 21st Century,
Beer Sheva, Israel, March, 1993.
\item{10.} C. Zhu and J.R. Klauder, ``The Classical Limit of Ultralocal
Scalar Fields,'' J. Math. Phys. (in press).
\item{11.} M. Rinke, Commun. math. Phys. {\bf 12}, 324 (1969); G.C.
Hegerfeldt, Commun. math. Phys. {\bf 45}, 137 (1975).
\bye
ENDBODY