Non-Archimedean Statistical Field Theory

Non-Archimedean Statistical Field Theory

W. A. Zúñiga-Galindo

University of Texas Rio Grande Valley, USA and Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México

INTRODUCTION

In arXiv:2006.05559, Non-Archimedean Statistical Field Theory

We construct (in a rigorous mathematical way) interacting quantum …eld theories over a p-adic spacetime in an arbitrary dimension.

We provide a large family of energy functionals E(ϕ, J)admitting natural

discretizations in …nite-dimensional vector spaces such that the partition function Zphys(J) = Z D(ϕ)e 1 KB TE(ϕ,J) ₍₁₎

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R

Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R_D

RD(ϕ)e

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R

Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R_D

RD(ϕ)e

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R_Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R_D

RD(ϕ)e

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R_Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R_D

RD(ϕ)e

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R_Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R

DRD(ϕ)e

1 KB TE(ϕ,J)

INTRODUCTION

Key fact: _DR=lim

!DlR= [∞l=1DRl ,DRl ,! D l+1

R . HereDRl is

…nite-dimensional real vector space. Given a test function ϕ_{2 DR}, there is l such that ϕ_{2 D}l

R, and thus ϕ has a natural discretization.

The key fact is not true for the real Schwartz space!

Discrete means that J _{2 D}l_R, and Z_lphys(J) =R_Dl

RD(ϕ)e

1 KB TE(ϕ,J)

In this case there is a cut-o¤, the support of the functions in _DRl is the ball with center at the origin and radius pl_.

Continuous means J _{2 DR}, and Zphys(J) =R

DRD(ϕ)e

INTRODUCTION

The goal of the work is to understand the limit:

R Dl RD(ϕ)e 1 KB TE(ϕ,J)_!R DRD(ϕ)e 1 KB TE(ϕ,J) _{as l !}∞.

Our main result is the construction of a measure on a function space such that Zphys(J) makes mathematical sense, and the calculations of the n-point correlation functions can be carried out using

perturbation expansions via functional derivatives, in a rigorous mathematical way.

Our results include ϕ4_{-theories. In this case, E(}

ϕ, J)can be

interpreted as a Landau-Ginzburg functional of a continuous Ising

model (i.e. ϕ₂R) with external magnetic …eld J.

If J =0, then E(ϕ, 0)is invariant under ϕ ! ϕ. We show that the

systems attached to discrete versions of E(ϕ, 0) have spontaneous

INTRODUCTION

The goal of the work is to understand the limit:

R Dl RD(ϕ)e 1 KB TE(ϕ,J)_!R DRD(ϕ)e 1 KB TE(ϕ,J) _{as l !}∞.

Our main result is the construction of a measure on a function space such that Zphys(J) makes mathematical sense, and the calculations of the n-point correlation functions can be carried out using

perturbation expansions via functional derivatives, in a rigorous mathematical way.

Our results include ϕ4_{-theories. In this case, E(}

ϕ, J)can be

interpreted as a Landau-Ginzburg functional of a continuous Ising

model (i.e. ϕ₂R) with external magnetic …eld J.

If J =0, then E(ϕ, 0)is invariant under ϕ ! ϕ. We show that the

systems attached to discrete versions of E(ϕ, 0) have spontaneous

INTRODUCTION

The goal of the work is to understand the limit:

R Dl RD(ϕ)e 1 KB TE(ϕ,J)_!R DRD(ϕ)e 1 KB TE(ϕ,J) _{as l !}∞.

Our main result is the construction of a measure on a function space such that Zphys(J) makes mathematical sense, and the calculations of the n-point correlation functions can be carried out using

perturbation expansions via functional derivatives, in a rigorous mathematical way.

Our results include ϕ4_{-theories. In this case, E(}

ϕ, J)can be

interpreted as a Landau-Ginzburg functional of a continuous Ising

model (i.e. ϕ₂R) with external magnetic …eld J.

If J =0, then E(ϕ, 0)is invariant under ϕ ! ϕ. We show that the

systems attached to discrete versions of E(ϕ, 0) have spontaneous

INTRODUCTION

The goal of the work is to understand the limit:

R Dl RD(ϕ)e 1 KB TE(ϕ,J)_!R DRD(ϕ)e 1 KB TE(ϕ,J) _{as l !}∞.

Our main result is the construction of a measure on a function space such that Zphys(J) makes mathematical sense, and the calculations of the n-point correlation functions can be carried out using

perturbation expansions via functional derivatives, in a rigorous mathematical way.

Our results include ϕ4_{-theories. In this case, E(}

ϕ, J)can be

interpreted as a Landau-Ginzburg functional of a continuous Ising

model (i.e. ϕ₂R) with external magnetic …eld J.

INTRODUCTION

whose symbol has a singularity at the origin.

The operator R

QN p

ϕ(x)W(∂, δ)ϕ(x)dNx is non local. Then E(ϕ, J)is a

INTRODUCTION

An important example of a W(∂, δ) operator is the Taibleson-Vladimirov

operator, which is de…ned as Dβ φ(x) = 1 p β 1 p β N Z QN p φ(x y) φ(x) ky_kβ+N p dNy _{= Fκ!}1_{x k}κkβ_pFx_!κφ , where β>0 and φ_{2 D} QNp .

If N = β=1, the energy functional

S(ϕ) =C RR Qp Qp ( ϕ(x) ϕ(y) jx y_jp )2 dxdy appears in p-adic string theory.

The W operators

Take R+:= fx 2R; x 0g, and …x a function wδ :Q

N

p !R+

satisfying: (i) w_δ(y)is a radial i.e. w_δ(y) =w_δ(ky_kp); (ii) there exist constants C0, C1 >0 and δ>N such that

C0kykδp wδ(kykp) C1kykδp, for y 2 QNp.

We now de…ne the operator W_δϕ(x) = Z QN p ϕ(x y) ϕ(x) w_δ(_ky_kp) dNy , for ϕ_{2 D} QNp .

Discretization of Energy Functionals

A discretization is obtained by considering truncations of p-adic numbers of the form

a lp l+a l+1p l+1+. . .+a0+. . .+al 1pl 1, for some l 1, i.e.

elements from Gl :=p lZNp/plZNp.

We denote by _Dl

R(QNp):= Dl_Rthe R-vector space of all test

functions of the form ϕ(x) = ∑

i₂Gl

ϕ(i)Ω plkx ik_p , ϕ(i)2R,

where i runs through a …xed system of representatives of Gl, and

Ω pl_{kx ik}

p is the characteristic function of the ball i+plZNp.

Notice that ϕ is supported on p l_ZN

p and that Dl_R is a …nite

dimensional vector space spanned by the basisnΩ pl_kx i_kp o

i₂Gl

. We identify ϕ_{2 D}l_R with the column vector [ϕ(i)]_i2G

Discretization of Energy Functionals

A discretization is obtained by considering truncations of p-adic numbers of the form

a lp l+a l+1p l+1+. . .+a0+. . .+al 1pl 1, for some l 1, i.e.

elements from Gl :=p lZNp/plZNp.

We denote by _Dl

R(QNp):= Dl_Rthe R-vector space of all test

functions of the form ϕ(x) = ∑

i2Gl

ϕ(i)Ω plkx ik_p , ϕ(i)2R,

where i runs through a …xed system of representatives of Gl, and

Ω pl_{kx ik}

p is the characteristic function of the ball i+plZNp.

Notice that ϕ is supported on p l_ZN

p and that Dl_R is a …nite

dimensional vector space spanned by the basisnΩ pl_kx i_kp o

i₂Gl

. We identify ϕ_{2 D}l_R with the column vector [ϕ(i)]_i2G

Discretization of Energy Functionals

A discretization is obtained by considering truncations of p-adic numbers of the form

a lp l+a l+1p l+1+. . .+a0+. . .+al 1pl 1, for some l 1, i.e.

elements from Gl :=p lZNp/plZNp.

We denote by _Dl

R(QNp):= Dl_Rthe R-vector space of all test

functions of the form ϕ(x) = ∑

i2Gl

ϕ(i)Ω plkx ik_p , ϕ(i)2R,

where i runs through a …xed system of representatives of Gl, and

Ω pl_{kx ik}

p is the characteristic function of the ball i+plZNp.

Notice that ϕ is supported on p l_ZN

p and that Dl_R is a …nite

dimensional vector space spanned by the basisnΩ pl_kx i_kp o

i2Gl

. We identify ϕ_{2 D}l_R with the column vector [ϕ(i)]_i2G

Discretization of Energy Functionals

If m is positive integer then ϕm(x) =

( ∑ i2Gl ϕ(i)Ω plkx ik_p )m = _∑ i2Gl ϕm(i)Ω plkx ik_p . The functional E_m0 (ϕ):= R QN p ϕm(x)dNx for m2Nr f0g, ϕ2 D_Rl , discretizes as E_m0 (ϕ) =p lN ∑ i₂Gl ϕm(i). E0(ϕ):= γ₄ RR QN p QNp fϕ(x) ϕ(y)g2 wδ(kx ykp) dN_xdN_y+α2 2 R QN p ϕ2(x)dNx 0.

The restriction of E0 to D_Rl (denoted as E₀(l)) provides a natural

Discretization of Energy Functionals

If m is positive integer then ϕm(x) =

( ∑ i2Gl ϕ(i)Ω plkx ik_p )m = _∑ i2Gl ϕm(i)Ω plkx ik_p . The functional E_m0 (ϕ):= R QN p ϕm(x)dNx for m2Nr f0g, ϕ2 D_Rl , discretizes as E_m0 (ϕ) =p lN ∑ i₂Gl ϕm(i). E0(ϕ):= γ₄ RR QN p QNp fϕ(x) ϕ(y)g2 wδ(kx ykp) dN_xdN_y+α2 2 R QN p ϕ2(x)dNx 0.

The restriction of E0 to D_Rl (denoted as E₀(l)) provides a natural

Discretization of Energy Functionals

If m is positive integer then ϕm(x) =

( ∑ i2Gl ϕ(i)Ω plkx ik_p )m = _∑ i2Gl ϕm(i)Ω plkx ik_p . The functional E_m0 (ϕ):= R QN p ϕm(x)dNx for m2Nr f0g, ϕ2 D_Rl , discretizes as E_m0 (ϕ) =p lN ∑ i₂Gl ϕm(i). E0(ϕ):= γ₄ RR QN p QNp fϕ(x) ϕ(y)g2 w_δ(kx y_kp) d N_xdN_y+α2 2 R QN p ϕ2(x)dNx 0.

The restriction of E0 to D_Rl (denoted as E₀(l)) provides a natural

Discretization of Energy Functionals

If m is positive integer then ϕm(x) =

( ∑ i2Gl ϕ(i)Ω plkx ik_p )m = _∑ i2Gl ϕm(i)Ω plkx ik_p . The functional E_m0 (ϕ):= R QN p ϕm(x)dNx for m2Nr f0g, ϕ2 D_Rl , discretizes as E_m0 (ϕ) =p lN ∑ i₂Gl ϕm(i). E0(ϕ):= γ₄ RR QN p QNp fϕ(x) ϕ(y)g2 w_δ(kx y_kp) d N_xdN_y+α2 2 R QN p ϕ2(x)dNx 0.

The restriction of E0 to D_Rl (denoted as E₀(l)) provides a natural

Discretization of Energy Functionals

We set U(l):=U = [Ui,j(l)]_i,j2Gl, where

Ui,j(l):= γ 2d(l , wδ) + α2 2 δi,j γ 2Ai,j(l), d(l , w_δ):= Z QN pnBNl dN_y w_δ(ky_kp) and Ai,j(l):= 8 > < > : p lN wδ(ki jkp) if i₆₌j 0 if i=j. . Lemma

Lizorkin spaces of second kind

The p-adic Lizorkin space of second kind ;

L:_{= L(}QNp) = 8 < :ϕ2 D(Q N p); R QN p ϕ(x)dNx =0 9 = ; LR:= LR(QN

p) = L(QNp) \ DR(QNp), the real version.

F L:_{= F L(}QN_p) =nbϕ2 D(QNp);bϕ(0) =0

o ,

The Fourier transform gives rise to an isomorphism of C-vector spaces from _Linto _{F L}.

The topological dual _L0 :_{= L}0(QN

p) of the spaceL is called the p-adic

Lizorkin spaces of second kind

The discrete p-adic Lizorkin space of second kind:

Ll :_{= L}l(QNp) = ( ϕ(x) =

∑

∑

l ₂N_{r f}0_g. The real version _Ll_R :_{= L}l_R(QNp) = Ll\ D_Rl .

Energy functionals in the momenta space

We use the alternative notation cϕ₁(j) =Re(bϕ(j)), bϕ₂(j) =Im(bϕ(j)).

Notice that F LlR= ( bϕ(κ) =

∑

and that the condition _bϕ(κ) =bϕ( κ) implies that bϕ₁( i) = bϕ₁(i)and

Energy functionals in the momenta space

We now de…ne the diagonal matrix B(r) =hB_i,j(r)i

i,j₂G_l+, r =1, 2, where B_i,j(r) := 8 < : γ 2Awδ(kjkp) + α2 2 if i=j 0 if i₆₌j.

Notice that B_i,j(1) =B_i,j(2). We set

B(l):=B(l , δ, γ, α2) =

B(1) 0

0 B(2).

The matrix B = [Bi,j]is a diagonal of size 2 #G_l+ 2 #G_l+ . In

Energy functionals in the momenta space

Lemma

Assume that α2>0. With the above notation the following formula holds

true: E₀(l)(ϕ):=E₀(l) ϕc₁(j),cϕ₂(j); j2G_l+ = " [cϕ₁(j)]_j2G+ l [cϕ₂(j)]_j2G+ l #T 2p lNB(l) " [cϕ₁(j)]_j2G+ l [cϕ₂(j)]_j2G+ l # 0,

Gaussian measures

Z(l) is a Gaussian integral, then

Gaussian measures

We de…ne the following family of Gaussian measures:

Gaussian measures

Thus for any Borel subset A of R(p2lN 1) ' FLl_R and any continuous and bounded function f :_{F L}l_R!R the integral

Gaussian measures

Lemma

There exists a probability measure space (X ,_F,P) and random variables " [cϕ₁(j)]_j2G+ l [cϕ₂(j)]_j2G+ l # , for l ₂N_{r f}0_g,

such that Pl is the joint probability distribution of

" [cϕ₁(j)]_j2G+ l [cϕ₂(j)]_j2G+ l # . The space (X ,_F,P) is unique up to isomorphisms of probability measure spaces. Furthermore, for any bounded continuous function f supported in

F LlR, we have R

f (bϕ)dPl(bϕ) =

R

Gaussian measures

For δ >N, γ, α2 >0, we de…ne the operator

Gaussian measures

We de…ne the distribution

G(x):=G(x; δ, γ, α2) = F_κ!1x 1 γ 2Awδ(kκkp) + α2 2 ! 2 D0 QNp .

By using the fact that γ 1

2Aw

δ(kκkp)+ α2

2

is radial and _{(F (F}ϕ))(κ) = ϕ( κ)

one veri…es that

Gaussian measures

Lemma

B is a positive, continuous bilinear form from DR QN

Gaussian measures

Corollary

The collection _fB_Y;_Y …nite dimensional subspace of _LRgis completely determined by the collection _fBl; l 2Nr f0gg. In the sense that given

any B_Y there is an integer l and a subset J G_l+, the case J = ∅ is

Gaussian measures in the non-Archimedean framework

The spaces

LR QNp ,!L2_R QNp ,! L0_R QNp

form a Gel’fand triple, that is,_LR QN_p is a nuclear space which is densely and continuously embedded in L2_R and _kg_k2_{2 = h}g , g_i for g _{2 LR} QN_p . The mapping

C : _LR QNp ! C

f _! e 12B(f ,f)

Gaussian measures in the non-Archimedean framework

By the Bochner-Minlos theorem, there exists a probability measure

P :=P(δ, γ, α2)called the canonical Gaussian measure on

L_R0 QN

p ,B , given by its characteristic functional as

R L0

R(QNp)

ep 1hW ,fidP(W) =e 12B(f ,f), f 2 LR QN

Gaussian measures in the non-Archimedean framework

The measure P is uniquely determined by the family of Gaussian measures

P_Y;_{Y LR}, …nite dimensional space ,

where P_Y(A) = 1 (2π)n2 R A e 12B(ψ,ψ)_{d ψ,} if _Y has dimension n.

Equivalently, P is uniquely determined by the family of bilinear forms

B_Y;_{Y LR}, …nite dimensional space ,

where B_Y denotes the restriction of the scalar product to B to _Y. Equivalently, P is uniquely determined by the family of bilinear forms

Gaussian measures in the non-Archimedean framework

Theorem

Assume that δ>N, γ>0, α2>0. (i) The cylinder probability measure P=P(δ, γ, α2) is uniquely determined by the sequence

Pl =Pl(δ, γ, α2), l 2Nr f0g, of Gaussian measures. (ii) Let

f : F LR QN

p !R be a continuous and bounded function. Then

lim l_!∞ R F LlR(QNp) f (bϕ)dPl(bϕ) = R F LR(QNp) f (bϕ)dP(bϕ).

Partition functions

We consider interactions of the form:

P(X) =a3X3+a4X4+. . .+a2kX2D 2R[X], with D 2,

satisfying _P(α) 0 for any α2R. Which implies that

exp α4

2

R

P(ϕ)dNx 1.

Each of these theories corresponds to a thermally ‡uctuating …eld which is de…ned by means of a functional integral representation of the partition function.

Partition functions

We consider interactions of the form:

P(X) =a3X3+a4X4+. . .+a2kX2D 2R[X], with D 2,

satisfying _P(α) 0 for any α2R. Which implies that

exp α4

2

R

P(ϕ)dNx 1.

Each of these theories corresponds to a thermally ‡uctuating …eld which is de…ned by means of a functional integral representation of the partition function.

Partition functions

We consider interactions of the form:

P(X) =a3X3+a4X4+. . .+a2kX2D 2R[X], with D 2,

satisfying _P(α) 0 for any α2R. Which implies that

exp α4

2

R

P(ϕ)dNx 1.

Each of these theories corresponds to a thermally ‡uctuating …eld which is de…ned by means of a functional integral representation of the partition function.

Partition functions

We assume that ϕ_{2 LR} QNp represents a …eld that performs thermal

‡uctuations. We also assume that in the normal phase the expectation value of the …eld ϕ is zero. Then the ‡uctuations take place around zero. The size of these ‡uctuations is controlled by the energy functional:

E(ϕ):=E0(ϕ) +Eint(ϕ), where Eint(ϕ):= α4 4 Z QN p P (ϕ(x))dNx, α4 0,

Partition functions

De…nition

Assume that δ>N, and γ, α2 >0. The free-partition function is de…ned

as

Z0 = Z0(δ, γ, α2) = Z

LR(QNp)

dP(ϕ).

The discrete free-partition function is de…ned as

Z0(l)= Z (l) 0 (δ, γ, α2) = Z Ll R(QNp) dPl(ϕ) for l ₂N_{r f}0_g.

Partition functions

De…nition

Assume that δ>N, and γ, α2, α4 >0. The partition function is de…ned

as Z = Z(δ, γ, α2, α4) = Z LR(QNp) e Eint(ϕ)_dP₍ ϕ).

The discrete partition functions are de…ned as

Partition functions

From a mathematical perspective a _{P (}ϕ)-theory is given by a cylinder

probability measure of the form 1_LR(ϕ)e Eint(ϕ)_dP Z LR(QNp) e Eint(ϕ)_dP = 1LR(ϕ)e Eint(ϕ)_dP Z (3)

in the space of …elds _LR QN_p . It is important to mention that we do not require the Wick regularization operation in e Eint(ϕ) _{because we are}

Partition functions

The m-point correlation functions of a …eld ϕ2 LR QN

p are de…ned as G(m)(x1, . . . , xm) = 1 Z Z LR(QNp) m

∏

The discrete m-point correlation functions of a …eld ϕ2 Ll

Generating functionals

We now introduce a current J(x_{) 2 LR} QN

p and add to the energy

functional E(ϕ)a linear interaction energy of this current with the …eld ϕ(x),

Esource(ϕ, J):= R

QN p

ϕ(x)J(x)dNx,

in this way we get a new energy functional

E(ϕ, J):=E(ϕ) +Esource(ϕ, J).

Notice that Esource(ϕ, J) = hϕ, Ji, whereh, idenotes the scalar

Generating functionals

De…nition

Assume that δ>N, and γ, α2, α4 >0. The partition function

corresponding to the energy functional E(ϕ, J) is de…ned as

Z(J; δ, γ, α2, α4):= Z(J) = 1 Z0 Z LR(QNp) e Eint(ϕ)+hϕ,Ji _dP,

and the discrete versions

Generating functionals

De…nition

For θ _{2 LR} QNp , the functional derivative DθZ(J)of Z(J)is de…ned as

Generating functionals

Lemma

Let θ1,. . . ,θm be test functions fromLR QNp . The functional derivative

D_θ1 D_θmZ(J)exists, and the following formula holds true: Dθ1 DθmZ(J) = 1 Z0 R LR(QNp) e Eint(ϕ)+hϕ,Ji _∏m i=1h ϕ, θii dP(ϕ).

Furthermore, the functional derivative D_θ1 D_θmZ(J) can be uniquely identi…ed with the distribution from _LR0 QN

Generating functionals

In an alternative way, one can de…ne the functional derivative δ

δJ(y)Z(J)

of _Z(J) as the distribution from _L0_R QN_p satisfying

R QN p θ(y) δ δJ(y)Z(J) (y)d N_{y =} d d eZ(J+eθ) _e=0. Using this notation, we obtain that

Generating functionals

Proposition

Free-…eld theory

denotes a normalization constant For J _{2 LR}, the equation

γ

2W (∂, δ) +

α2

2 ϕ0 =J

has unique solution ϕ_{0 2 LR}. Indeed, _cϕ₀(κ) = γ bJ(κ)

2Aw_δ(kκkp)+α22

is a test function satisfying _cϕ₀(0) =0. Furthermore,

ϕ (x) =F _!1x(

1

Free-…eld theory

Proof.

We now change variables in _Z0(J) as ϕ= ϕ₀+ϕ0,

Free-…eld theory

The correlation functions G₀(m)(x1, . . . , xm)of the free-…eld theory are

obtained from the functional derivatives of _Z0(J)at J =0:

Perturbation expansions for phi-4-theories

The existence of a convergent power series expansion for Z(J) (the perturbation expansion) in the coupling parameter α4 follows from the fact

that exp( Eint(ϕ) + hϕ, Ji)is an integrable function, by using the

dominated convergence theorem, more precisely, we have

Perturbation expansions for phi-4-theories

Theorem

Assume that _P(ϕ) = ϕ4. The n-point correlation function of the …eld ϕ

admits the following convergent power series in the coupling constant:

Ginzburg-Landau phenomenology

A non-Archimedean Ginzburg-Landau free energy:

E(ϕ, J): E(ϕ, J ; δ, γ, α2, α4) = γ(T) 2 RR QN p QNp fϕ(x) ϕ(y)g2 w_{δ k}x y_kp dNxdNy +α2(T) 2 Z QN p ϕ2(x)dNx+ α4(T) 4 Z QN p ϕ4(x)dNx Z QN p ϕ(x)J(x)dNx, where J, ϕ2 DR, and γ(T) = γ+O((T Tc)); α2(T) = (T Tc) +O((T Tc)2); α4(T) = α4+O((T Tc)),

Ginzburg-Landau phenomenology

We consider that ϕ_{2 D}l

Ris the local order parameter of a

continuous Ising system with ‘external magnetic …eld’J _{2 DR}l .

The system is contained in the ball B_lN. We divide this ball into sub-balls (boxes) BN_l(i), i₂Gl. The volume of each of these balls is

p lN and the radius is a :=p l.

Each ϕ(i)2 R represents the ‘average magnetization’in the ball

BN_l(i). We take ϕ(x) =_∑i2Gl ϕ(i)Ω plkx ik_p which is a

locally constant function.

Ginzburg-Landau phenomenology

We consider that ϕ_{2 D}l

Ris the local order parameter of a

continuous Ising system with ‘external magnetic …eld’J _{2 DR}l .

The system is contained in the ball B_lN. We divide this ball into sub-balls (boxes) BN

l(i), i2Gl. The volume of each of these balls is

p lN and the radius is a := p l.

Each ϕ(i)2 R represents the ‘average magnetization’in the ball

BN_l(i). We take ϕ(x) =_∑i2Gl ϕ(i)Ω plkx ik_p which is a

locally constant function.

Ginzburg-Landau phenomenology

We consider that ϕ_{2 D}l

Ris the local order parameter of a

continuous Ising system with ‘external magnetic …eld’J _{2 DR}l .

The system is contained in the ball B_lN. We divide this ball into sub-balls (boxes) BN

l(i), i2Gl. The volume of each of these balls is

p lN and the radius is a := p l.

Each ϕ(i)2 R represents the ‘average magnetization’in the ball

BN_l(i). We take ϕ(x) =_∑i2Gl ϕ(i)Ω plkx ik_p which is a

locally constant function.

Ginzburg-Landau phenomenology

We consider that ϕ_{2 D}l

Ris the local order parameter of a

continuous Ising system with ‘external magnetic …eld’J _{2 DR}l .

The system is contained in the ball B_lN. We divide this ball into sub-balls (boxes) BN

l(i), i2Gl. The volume of each of these balls is

p lN and the radius is a := p l.

Each ϕ(i)2 R represents the ‘average magnetization’in the ball

BN_l(i). We take ϕ(x) =_∑i2Gl ϕ(i)Ω plkx ik_p which is a

locally constant function.

Ginzburg-Landau phenomenology

Then considering ϕ(i)₂R as the continuous spin at the site i₂Gl, the

partition function of our continuos Ising model is

Z(l)(β) =

∑

fϕ(i);i2Glg

e βE(ϕ(i),J(i))_.

Theorem

The minimizers of the functional E(ϕ, 0), ϕ2 D_Rl are constant solutions

Spontaneous symmetry breaking

If J =0, the …eld ϕ_{2 D}l_Ris a minimum of the energy functional E , if it satis…es (5). When T >TC we have α2 >0 and the ground state is ϕ₀ =0. In contrast, when T <TC, α2 <0, there is a degenerate ground

state ϕ₀ with

ϕ₀ =

r

α2 α4.

This implies that below TC the systems must pick one of the two states

+ϕ₀ or ϕ₀, which means that there is a spontaneous symmetry