Ultraviolet Stability for QED in $d=3$

Abstract

We continue the study of the ultraviolet problem for QED in $d=3$ using Balaban’s formulation of the renormalization group. The model is defined on a fine toroidal lattice and we seek control as the lattice spacing goes to zero. Drawing on earlier papers in the series the renormalization group flow is completely controlled for weak coupling. The main result is an ultraviolet stability bound in a fixed finite volume.

Appendices

Notation

This is a guide to the notation used in the text, with references to exact definitions. Everything refers to decreasing sequence of small field regions $\Omega _1 \supset \Lambda _1 \supset \Omega _2 \supset \Lambda _2 \supset \cdots \supset \Omega _k \supset \Lambda _k$.

• 1. Gauge fields

• After k steps one has fundamental gauge fields $(A_0, A_1, \cdots , A_k)$ with the final field $A_k$ on a unit lattice and previous fields $A_j$ scaled down from a unit lattice by $L^{-(k-j)}$. One is particularly interested in fields $A_{k, \mathbf {\Omega }}= ( A_{0, \Omega _1^c},A_{1, \delta \Omega _1}, \cdots , A_{k-1, \delta \Omega _{k-1}}, A_{k, \Omega _k} )$ localized in $\delta \Omega _j = \Omega _j - \Omega _{j+1}$ which play a role in every subsequent step.

• ${{\mathcal {Q}}}_{k, \mathbf {\Omega }}$ is a multiscale averaging operator defined in (218) in [31]. The field on ${{\mathbb {T}}}^{-k}_{N -k}$ is the minimizer of subject to and a gauge condition [31]. This could be either Landau gauge or axial gauge with notation reserved for the Landau gauge and the axial gauge denoted . They are linear functions of the fundamental fields.

• There are also a single step minimizers $A^{\min } _{k, \mathbf {\Omega }^+ }$ on ${{\mathbb {T}}}^0_{N -k}$ depending on $\mathbf {\Omega }^+ = ( \mathbf {\Omega }, \Omega _{k+1} )$. It is defined as the minimizer of in $A_{k, \mathbf {\Omega }} =A_k$ on $\Omega _{k+1}$ subject to ${{\mathcal {Q}}}A_k = A_{k+1}$ and an axial gauge condition, with $A_{ k, \mathbf {\Omega }}$ fixed on $\Omega ^c_{k+1}$. See (241) in [31].

• $\mathbf {\Omega }^+ = ( \mathbf {\Omega }, \Omega _{k+1} )$ and on ${{\mathbb {T}}}^{-k}_{N -k}$ is the field before scaling to ${{\mathbb {T}}}^{-k-1}_{N-k-1}$. We have that is gauge equivalent to ${{\mathcal {H}}}_{k, \mathbf {\Omega }} A^{\min }_{k, \mathbf {\Omega }^+ }$.

• Fluctuation fields $Z_k$ on $\Omega _{k+1}$ are defined by $A_{k, \mathbf {\Omega }} = A^{\min } _{k, \mathbf {\Omega }^+ } + Z_k$. Then is gauge equivalent to where ${{\mathcal {Z}}}_{k, \mathbf {\Omega }} = {{\mathcal {H}}}_{k, \mathbf {\Omega }} Z_k$. The $Z_k$ satisfy an axial gauge condition and are parametrized by $Z_k = C {\tilde{Z}}_k$, see [28, 31].

• and $A^{\min }_{k, \mathbf {\Omega }^+ (\square ) }$ are versions of the above operators approximately localized in M-cubes $\square$. They are defined in (52),(56).

• 2. Fermi fields

• After k steps we have fundamental fermi fields $(\Psi _0, \Psi _1, \cdots , \Psi _k )$ with the final field $\Psi _k$ on a unit lattice and previous fields $\Psi _j$ scaled down from a unit lattice by $L^{-(k-j)}$. We are particularly interested in fields $\Psi _{k, \mathbf {\Omega }}= ( \Psi _{0, \Omega _1^c},\Psi _{1, \delta \Omega _1}, \cdots , \Psi _{k-1, \delta \Omega _{k-1}}, \Psi _{k, \Omega _k} )$.

• is a multiscale covariant averaging operator defined in (39) in [31]. The field is a critical point of the action

  

  (526)

  for some constants ${\mathbf {b}}^{(k)}$, see (51)-(54) in [31]. It is linear in $\Psi _{k, \mathbf {\Omega }}$.

• There are also single step critical points , defined as the critical point in $\Psi _{k, \mathbf {\Omega }}= \Psi _k$ on $\Omega _{k+1}$ of

  

  (527)

  with $\Psi _{k, \mathbf {\Omega }}$ fixed on $\Omega _{k+1}^c$. See lemma 3 in [31].

• on ${{\mathbb {T}}}^{-k}_{N -k}$ is the field before scaling to ${{\mathbb {T}}}^{-k-1}_{N-k-1}$. Then , see lemma 3 in [31]

• Fluctuation fields $W_k$ are defined on $\Omega _{k+1}$ by . Then where , see (71)-(73) in [31].

• 3. Miscellaneous.

• A polymer X is a connected union of M-cubes, and $|X|_M$ is the number of cubes. The quantity $d_M(X)$ is defined by specifying that $Md_M(X)$ is the length of a shortest continuum tree joining the cubes in X.

• If X is specified as a union of M cubes or $Mr_k$ cubes, then ${\tilde{X}}$ is X enlarged by a layer of cubes of the same type, and $X^{\natural } \equiv ((X^c)^{\sim } )^c$ is X shrunk by a layer of cubes.

• ${{\mathcal {R}}}_{k, \mathbf {\Omega }}, {\tilde{{{\mathcal {R}}}}}_{k, \mathbf {\Omega }}$ are analyticity regions for a gauge field defined in (36).

• In passing from step k to step $k+1$ four new characteristic functions are introduced. They are $\chi _{k+1}$ defined in (50) which limits the size of fields . This is followed by $\chi '_k$ defined in (55) which limits the size of the fluctuation fields $A_k - A^{\min } _{k, \mathbf {\Omega }(\square ) }$. This is not quite $Z_k$, but this is corrected by $\chi ^{\dagger }_k$ defined in (104) which gives local bounds on $Z_k$. The fluctuation integral is changed to unit covariance which delocalizes the dependence on $Z_k$; the final characteristic function ${\hat{\chi }}_k$ defined in (150) restores local bounds on $Z_k$.

• Expansions in $\chi _{k+1}, \chi _k'$ define $\Omega _{k+1}$. Expansions in $\chi _k^{\dagger }, {\hat{\chi }}_k$ define $\Lambda _{k+1}$.

Norms

The effective actions in our renormalization group analysis will be expressed in terms of polymer functions which are elements of a Grassmann algebra. We define some norms on such elements. These can also be combined for mixed versions.

1.1 Single scale

Consider the unit lattice, say ${{\mathbb {T}}}^0_{N -k}$. Fermi fields $\Psi ({\mathsf {x}})$ are the generators of a Grassmann algebra indexed by ${\mathsf {x}}= (x, \beta , \omega )$ with $x \in {{\mathbb {T}}}^0_{N -k}$, $1 \le \beta \le 4$, and $\omega =0,1$ and have the form $\Psi ( x, \beta , 0 ) = \Psi _{\beta }( x)$ and $\Psi ( x, \beta , 1 ) = {\bar{\Psi }}_{k,\beta }( x)$. We consider elements of the Grassmann algebra of the form

$$\begin{aligned} E(\Psi ) = \sum _{n=0}^{\infty } \frac{1}{n!} \sum _{{\mathsf {x}}_1, \dots , {\mathsf {x}}_n } E_n( {\mathsf {x}}_1, \dots , {\mathsf {x}}_n) \Psi ( {\mathsf {x}}_1 ) \cdots \Psi ( {\mathsf {x}}_n) \end{aligned}$$

(528)

This is actually a finite sum since ${{\mathbb {T}}}^0_{N -k}$ is finite. A norm with a parameter $h>0$ is defined by

$$\begin{aligned} \Vert E \Vert _h = \sum _{n=0}^{\infty } \frac{ h^n }{n!} \sum _{ {\mathsf {x}}_1, \dots , {\mathsf {x}}_n } | E_n( {\mathsf {x}}_1, \dots , {\mathsf {x}}_n) | \end{aligned}$$

(529)

1.2 Dressed fields

Now consider a fine lattice, say ${{\mathbb {T}}}^{-k}_{N -k}$. Fermi fields $\psi _k (\xi )$ are elements of certain Grassmann algebras. indexed by $\xi = (x, \beta , \omega )$ with $x \in {{\mathbb {T}}}^{-k}_{N -k}$, $1 \le \beta \le 4$, and $\omega =0,1$ and have the form $\psi (x, \beta , 0) = \psi _{\beta } (x)$ and $\psi (x, \beta , 1) = {\bar{\psi }}_{\beta } (x)$. We have in mind smeared functions of the fundamental fields like . We consider elements of the Grassmann algebra of the form

$$\begin{aligned} \begin{aligned} E( \psi ) =&\sum _{n=0}^{\infty } \frac{1}{n!} \int \ E_{n}( \xi _1, \dots , \xi _n) \psi ( \xi _1 ) \cdots \psi ( \xi _n) d\xi _1 \cdots d\xi _n \\ \end{aligned} \end{aligned}$$

(530)

Here with $\eta = L^{-k}$ we define $\int d \xi = \sum _{x,\beta , \omega } \eta ^3$. A norm on the kernel is defined by

$$\begin{aligned} \Vert E_{n} \Vert _h = \sum _{n=0}^{\infty } \frac{ h^n }{n!} \int | E_{n}( \xi _1, \dots , \xi _n) | d \xi _1 \cdots d \xi _n \end{aligned}$$

(531)

A further variation allows inclusion of the Holder derivative $\delta _{\alpha } \psi (x,y)$ (defined for $|x-y| <1$) as a separate field. For $\zeta =(x,y, \beta , \omega )$ we define $\delta _{\alpha } \psi (\zeta )$ by $\delta _{\alpha } \psi (x,y, \beta , 0 ) = \delta _{\alpha } \psi _{\beta } (x,y)$ and $\delta _{\alpha } \psi (x,y, \beta , 1 ) = \delta _{\alpha } {\bar{\psi }}_{\beta } (x,y)$. An integral over $\zeta$ is $\int d \zeta = \sum _{|x-y|<1,\beta , \omega } \eta ^6$. We consider functions of the form

$$\begin{aligned} \begin{aligned}&E( \psi _k, \delta _{\alpha } \psi _k ) = \sum _{n,m=0}^{\infty } \frac{1}{n!m!} \int E_{nm}( \xi _1, \dots , \xi _n, \zeta _1, \dots , \zeta _m) \\&\quad \quad \psi (\xi _1 ) \cdots \psi ( \xi _n) \ \delta _{\alpha } \psi ( \zeta _1 ) \cdots \delta _{\alpha } \psi ( \zeta _m) \ d\xi _1 \cdots d\xi _n d\zeta _1 \cdots d\zeta _m \end{aligned} \end{aligned}$$

(532)

Norms for the kernels are defined for a pair of parameters ${\mathbf {h}}= (h_1, h_2)$ by

$$\begin{aligned} \Vert E \Vert _{ {\mathbf {h}}} = \sum _{n,m=0}^{\infty } \frac{ h_1^n h_2^m }{n!m!} \int | E_{nm}( \xi _1, \dots , \xi _n, \zeta _1, \dots , \zeta _m) | d \xi _1 \cdots d \xi _n d \zeta _1 \cdots d \zeta _m \end{aligned}$$

(533)

1.3 Multiscale

Now we consider a multiscale version . As in the text suppose we are given a decreasing sequence of small field regions $\mathbf {\Omega }= (\Omega _1, \Omega _2, \dots , \Omega _k)$ and fermi fields $\Psi _{k, \mathbf {\Omega }} = ( \Psi _{1, \delta \Omega _1}, \dots , \Psi _{k-1, \delta \Omega _{k-1}}, \Psi _{k, \delta \Omega _k})$ with $\delta \Omega _k = \Omega _k$. The fields are the generators of a Grassmann algebra indexed by ${\mathsf {x}}= (x, \beta , \omega )$ as in Sect. B.1 except that now $\Psi _{j, \delta \Omega _j} ({\mathsf {x}})$ has $x \in \delta \Omega _{j}^{(j)} \subset {{\mathbb {T}}}^{-(k-j)}_{N-k}$. We consider elements of the form

$$\begin{aligned} E( \Psi _{k,\mathbf {\Omega }} ) = \sum _{n_1, \cdots , n_k}^{\infty } \frac{1}{n_1! \cdots n_k! } \int E( \underline{ {\mathsf {x}}_1}, \dots , \underline{{\mathsf {x}}_k} ) \Psi _{1, \delta \Omega _1}(\underline{ {\mathsf {x}}_1 ) } \cdots \Psi _{k, \delta \Omega _k}( \underline{{\mathsf {x}}_n} ) \ d\underline{{\mathsf {x}}_1} \cdots , d \underline{{\mathsf {x}}_n } \end{aligned}$$

(534)

where for $\underline{ {\mathsf {x}}_j } =( {\mathsf {x}}_{ j,1} \cdots {\mathsf {x}}_{j,n_j} )$

$$\begin{aligned} \Psi _{j, \delta \Omega _j}(\underline{ {\mathsf {x}}_j ) } = \Psi _j ( {\mathsf {x}}_{ j,1}) \cdots \Psi _j( {\mathsf {x}}_{j,n_j} ) \end{aligned}$$

(535)

and where $\int d {\mathsf {x}}_i = \sum _{x_i} L^{-3(k-i)}$. The norms on the kernels with a multiweight ${\mathbf {h}}= (h_1, \dots , h_k)$ are

$$\begin{aligned} \Vert E \Vert _ {{\mathbf {h}}} = \sum _{n_1, \cdots , n_k}^{\infty } \frac{h_1^{n_1} \cdots h_k^{n_k} }{n_1!\cdots n_k!} \int | E( \underline{ {\mathsf {x}}_1}, \dots , \underline{{\mathsf {x}}_k} ) | \ d\underline{{\mathsf {x}}_1} \cdots , d \underline{{\mathsf {x}}_n } \end{aligned}$$

(536)

Bound on

We study the operator which is an approximate left inverse of the minimizer ${{\mathcal {H}}}_{k, \mathbf {\Omega }}$ and is defined on functions on $\Omega _1 \subset {{\mathbb {T}}}^{-k}_{N -k}$ by

(537)

Lemma 28

For there is a constant C such that

(538)

Proof

We have

(539)

The last two terms satisfy the bound so it suffices to consider . On $\Omega _k$ this is and in [29, 30] we show that

(540)

where last step follows since on $\Omega _k$.

On $\delta \Omega _j$ the operator is We treat this by scaling it to the previous result. We have

(541)

where we used . The bracketed expression is now the same as the previous case but on ${{\mathbb {T}}}^{-j}_{N-j}$ rather than ${{\mathbb {T}}}^{-k}_{N -k}$. We have therefore

(542)

where the last step follows since on $\delta \Omega _j$. $\square$

More bounds

We give a bound relating and the fundamental fields $A_j$. First a preliminary estimate.

Lemma 29

Let X be a union of unit blocks in ${{\mathbb {T}}}^{-k}_{N -k}$ with ${\tilde{X}}$ an enlargement by a layer of unit cubes. For a function F on plaquettes in ${{\mathbb {T}}}^{-k}_{N -k}$ the $L^2$ norms satisfy

$$\begin{aligned} \Vert {{\mathcal {Q}}}_k^{(2)} F \Vert _X \le {{\mathcal {O}}}(1)\Vert F \Vert _{ {\tilde{X}} } \end{aligned}$$

(543)

Proof

For $x \in {{\mathbb {T}}}^{-k}_{N -k}$ let $P_x = [ x, x+ e_{\mu }, x+ e_{\mu } + e_{\nu }, x + e_{\nu } ]$. Then for $y \in {{\mathbb {T}}}^0_{N -k}$ we have

$$\begin{aligned} ({{\mathcal {Q}}}_k^{(2)} F )(P_y) = \int _{|x-y| \le \frac{1}{2}} L^{-2k} \sum _{p \in P_x} F(p) = \int _{|x-y| \le \frac{1}{2}} \int _{p \in P_x} F(p) \end{aligned}$$

(544)

By the Schwarz inequality in the p integral

$$\begin{aligned} \Big | \int _{p \in P_x} F(p) \Big |^2 \le \int _{p \in P_x} |F(p)|^2 \end{aligned}$$

(545)

and then by the Schwarz inequality in the x integral

$$\begin{aligned} | ({{\mathcal {Q}}}_k^{(2)} F )(P_y) |^2 \le \int _{|x-y| \le \frac{1}{2}} \int _{p \in P_x} |F(p)|^2 \end{aligned}$$

(546)

Then

$$\begin{aligned} \begin{aligned} \Vert {{\mathcal {Q}}}_k^{(2)} F \Vert ^2_X&\equiv \sum _{y:P_y \cap X \ne \emptyset } | ({{\mathcal {Q}}}_k^{(2)} F )(P_y) |^2 \le \int _{x \in {\tilde{X}}} \int _{p \in P_x} |F(p)|^2 \\&= \int _{p \in {\tilde{X}}} \int _{x\in {\tilde{X}}: P_x \ni p} |F(p)|^2 \le {{\mathcal {O}}}(1)\Vert F \Vert ^2_{{\tilde{X}}} \end{aligned} \end{aligned}$$

(547)

This completes the proof. $\square$

Now consider which satisfies . Then

(548)

In $\Omega _k$ this says . It follows by lemma  29 that

(549)

We want to drop the restriction to $\Omega _k$ here.

Lemma 30

Let $X = \cup X_j$ where $X_j \subset \delta \Omega _j$ is a union of $L^{-(k-j)}$ cubes whose enlargements at that scale also satisfy ${\tilde{X}}_j \subset \delta \Omega _j$. Then

(550)

Proof

$A_j$ is a function on $\delta \Omega ^{(j)}_j \subset {{\mathbb {T}}}^{-(k-j)}_{N-k}$. Therefore $A_{j,L^{k-j}}$ is a function on a subset of the unit lattice ${{\mathbb {T}}}^{0}_{N-j}$ and on $L^k \delta \Omega _j$ we have . Then by a bound like (549)

(551)

Summing over j gives the result. $\square$