Interplay between reflection positivity and crossing symmetry in the bootstrap approach to CFT

The characterization and the properties of positive definite functions of a single variable have been intensely studied in the early twentieth century, mainly by Schoenberg, Widder and Bernstein. Several known properties of the vacuum correlation functions in relativistic QFT are obtained by the applications of some of these results. In this section, we give some definitions and enunciate key theorems on positive definite functions that are relevant to our paper. For a complete study of these topics we refer the readers to Bernstein ; Widder ; Berg (also see 2016arXiv160804010J for a brief account of the main theorems).

There are two notions of positive definiteness for a function of a single variable. The definition we use here is the following:

Positive definiteness in this sense turns out to be a very restrictive condition. A surprising consequence of this property is the following: if $f:(a,b)\rightarrow{\mathbb{R}}$ is positive definite and continuous in $(a,b)$, then it is $C^{\infty}(a,b)$ (even more, it is real analytic there Widder ).

Moreover, the derivatives $f^{n}(x)$ of order $n$ satisfy an infinite set of inequalities valid at any $x\in(a,b)$: the $N\times{N}$ matrices ${H}^{(N,f)}$ of coefficients $\left({H^{(N,f)}}\right)_{m,n}=f^{(n+m)}$ ($n,m=0,...,N-1$) are positive definite,

$$\det H^{(N,f)}=\left|\begin{array}[]{ccccc}f&f^{(0+1)}&f^{(0+2)}&..&f^{(0+N-1)}\\ f^{(1+0)}&f^{(1+1)}&.&.&..\\ ..&..&.&.&\\ f^{(N-1+0)}&.&.&.&f^{(N-1+N-1)}\\ \end{array}\right|\geq{0}$$

(4)

for all $N\in{\mathbb{N}}$. Conversely, an analytic function satisfying this infinite set of inequalities is pd. In fact, the inequalities (4) need only be satisfied at one point in $(a,b)$ and then they are automatically satisfied throughout the interval 2016arXiv160804010J .

An obvious consequence of the definition of positive definiteness is that a pd function is non-negative. A less obvious consequence is that the even derivatives of a pd function are also pd (this follows easily from the inequalities (4)), and hence non-negative. Note also from the definition that a linear combination of pd functions with positive coefficients is also pd.

Simple examples of pd functions are $f(t)=e^{\lambda{t}}$ for $\lambda$ a real number. The positive definiteness can be checked easily both from the definition and from the inequalities (4). The definition of pd function requires that $\sum_{i,j=1..N}c_{i}c_{j}f(\frac{t_{i}+t_{j}}{2})\geq{0}$. In this case, $\sum_{i,j=1..N}c_{i}c_{j}f(\frac{t_{i}+t_{j}}{2})=(\sum_{i=1...N}c_{i}e^{\frac{\lambda{t}_{i}}{2}})^{2}\geq{0}$. Therefore, $f$ is pd. Checking the inequalities is trivial since all the determinants are just $0$. Linear combinations of exponential functions with positive coefficients will also be pd. In particular, a constant function $f(t)=c$, with $c\geq{0}$ is a pd function.

Note that the exponential $f(t)=e^{\lambda{t}}$ is AM for $\lambda>0$ and CM for $\lambda<0$. A pd function can always be written as the sum of an AM function and a CM function. This follows from a classical theorem (see Widder ,Theorem VI.21), which states that a function $f$ on $(a,b)$ is pd if and only if it admits the following integral representation:

$$f(t)=\int_{-\infty}^{\infty}e^{-\lambda{t}}g(\lambda)d\lambda=\int_{-\infty}^{0}e^{-\lambda{t}}g(\lambda)d\lambda+{\int_{0}}^{\infty}e^{-\lambda{t}}g(\lambda)d\lambda$$

(5)

where $g$ is non-negative (strictly speaking, $g(\lambda)d\lambda$ has to be understood as a Borel measure). Note that the first term on the rhs side above is AM and the second term is CM.

Most important for this paper are the pd functions defined on $(0,+\infty)$ (or more generally on any interval of the form $(a,+\infty)$) which are bounded at infinity. From the decomposition (5) it follows that such functions are necessarily CM.

Roughly speaking, this is because the first term in (5) diverges as $t\to\infty$, so this term must be absent in order for $f$ to be bounded at infinity (for a technical proof of this see Bernstein ). Conversely, it can be shown Widder that any CM function on $(0,+\infty)$ admits the integral representation of the second term in (5), and hence it is pd. In other words, the space of pd functions on $(0,+\infty)$ which are bounded at infinity is equal to the space of CM functions on the same interval.

This equivalence gives rise to additional inequalities to the ones given by equation (4), which come from the obvious fact that, if $f$ is CM, then $-f^{\prime}$ is also CM. Using this and the above equivalence, we conclude that, for $f$ pd on $(0,+\infty)$ and bounded at infinity, $-f^{\prime}$ is also pd.

The additional inequalities arise from substituting $f$ by $-f^{\prime}$ in (4):

$$\det H^{(N,-f^{\prime})}=(-1)^{N}\left|\begin{array}[]{ccccc}f^{\prime}&f^{\prime(0+1)}&f^{\prime(0+2)}&..&f^{\prime(0+N-1)}\\ f^{\prime(1+0)}&f^{\prime(1+1)}&.&.&..\\ ..&..&.&.&\\ f^{\prime(N-1+0)}&.&.&.&f^{\prime(N-1+N-1)}\\ \end{array}\right|\geq{0}$$

(6)

Thus, pd functions on $(0,+\infty)$ which are bounded at infinity are characterized by two equivalent sets of conditions: (i) $(-1)^{n}f^{(n)}\geq{0}$ and (ii) equations (4) and (6). The first set of conditions appears to be much simpler than the second, but the second is more useful in some cases.

Having these tools in mind, let us come back to the issue of RP. What we will do next is to impose further restriction to the sequences $X_{i}$ mentioned in the previous subsection.

Let us consider a sequence of aligned points in a euclidean $d$-dimensional space, in the direction of the coordinate “$t$”. Then, we can write: $X_{i}=\{t^{i}_{1},t^{i}_{2},...t^{i}_{N}\}$, with the time coordinate ordered as $0<t_{\alpha}<t_{\alpha+1}$. The main idea is to take a family of sequences $\{X_{i}\}$ such that, given a single real parameter $\lambda(i)$ that defines univocally the sequences, the quantity $S_{2N}(\Theta(X_{j})X_{i})$ will then be a function of a “sum variable” $\lambda=\lambda(i)+\lambda(j)$. If that is possible for the chosen family, then we can apply the machinery for a positive definite function of a single variable $S_{2N}(\lambda)$ reviewed above. Restricting ourselves to length 2 sequences, there are two natural families (see Figure 1):

1. 1.

   First family ($\eta-$type): $X_{i}=\{t^{i},t^{i}+L\}$, $t^{i}>0,L>0$.

2. 2.

   Second Family ($\rho-$type): $X_{i}=\{d_{0},d_{0}+L_{i}\}$, $d_{0}>0,L_{i}>0$

The names refers to the parameters that will be introduced soon. In the first one, the two points of each sequence are separated by a fixed quantity $L$, and the single real variable that defines each sequence in this family is $t^{i}$.

In the second one, the separation is not fixed but the closest point to the origin is the same: $d_{0}$. The one parameter that identifies each sequence is $L_{i}$.

As we said before, one can consider that these two families correspond (formally) to a special choice of sharp test function. Let us see how the additive property holds in each family.

In the first one, it is enough to use invariance under time translation:

	$\displaystyle S_{4}(\Theta(X_{j})X_{i})=S_{4}(-t^{j}_{2},-t^{j}_{1},t^{i}_{1},t^{i}_{2})=S_{4}(0,t^{j}_{2}-t^{j}_{1},t^{j}_{2}+t^{i}_{1},t^{j}_{2}+t^{i}_{2})=$
	$\displaystyle S_{4}(0,L,t^{j}_{1}+t^{i}_{1}+L,t^{j}_{1}+t^{i}_{1}+2L)=F(t^{i}_{1}+t^{j}_{1})$		(7)

In this case, the Schwinger functions appearing in the RP inequalities are functions of the sum of the starting points of each sequence, since $L$ is a fixed parameter. It is important to mention here that the first family has this property in any QFT, not just in CFT.

For the second family, it is not immediate to identify the pd function of the sum of the parameters. In fact, in a general QFT there is not such function. However, we will see in the next section that in CFT the conformal invariance will allow us to rewrite the 4-point function $S_{4}(\Theta(X_{j})X_{i})$ as a function of the sum of the parameter $\log(\frac{2d_{0}}{L_{i}}+1)$.

In the $\rho-$family, it is not obvious that $S_{4}(\Theta(X_{j})X_{i})$ becomes a function of the sum of a suitable parameter of $X_{j}$ and $X_{i}$. In order to see that, let us define the parameter $\rho_{i}=\log(\frac{2d_{0}}{L_{i}}+1)$. The cross-ratio $x$ associated with the 4 points in the sequence $\Theta(X_{j})X_{i})$ becomes then $x=e^{-(\rho_{i}+\rho_{j})}=e^{-\rho}$, with $\rho$ the sum variable. Evaluating (8) in this particular case, we find:

	$\displaystyle S_{4}(x_{1},x_{2},x_{3},x_{4})$	$\displaystyle=S_{4}(-t^{j}_{2},-t^{j}_{1},t^{i}_{1},t^{i}_{2})$		(10)
		$\displaystyle=\left(\frac{(1-x)^{2}}{x}\right)^{\Delta_{21}}(2d_{0})^{-2\Delta_{21}}(L_{i}{L_{j}})^{-2\Delta_{1}}g^{21}_{21}(x)$

The positive definiteness requires that $\sum_{i,j}C_{i}{C^{*}}_{j}S_{4}(-t^{j}_{2},-{t^{j}}_{1},{t^{i}}_{1},{t^{i}}_{2})\geq{0}$. Since the $C_{i}$ are arbitrary numbers, and $x$ is the product $e^{-\rho_{i}}e^{-\rho_{j}}$, one can redefine

$$\tilde{C}_{i}=\left(\frac{e^{\rho_{i}}}{2d_{0}}\right)^{\Delta_{21}}{L_{i}}^{-2\Delta_{1}}C_{i}$$

and conclude that

$$(1-x(\rho))^{2\Delta_{21}}g^{21}_{21}(x(\rho))$$

(11)

is pd as a function of $\rho=-\log(x)$. Moreover, since it is bounded as $\rho\rightarrow{\infty}$, it has to be a CM function. This is equivalent to say that it should have a positive Laplace anti-transform and then the coefficients $b_{n}$ in the formal expansion:

$$(1-x)^{2\Delta_{21}}g^{21}_{21}(x)=\sum_{n}b_{n}x^{n}=\sum_{n}b_{n}e^{-n\rho}$$

(12)

should be all non-negative (where $n$ denotes a generic non-integer power). This coefficients are not identical to the ones of $g$, since there is a prefactor $(1-x)^{2\Delta_{21}}$. If we write formally $g^{21}_{21}(x)=\sum_{m}a_{m}x^{m}$ and insert the expansion:

$$(1-x)^{2\Delta_{21}}=\sum_{l=0}^{\infty}(-2\Delta_{21})_{l}\frac{x^{l}}{l!}$$

where $(-2\Delta_{21})_{l}$ is the Pochhammer symbol, the previous positivity conditions implies that the coefficients $b_{n}$:

$$b_{n}\equiv\sum_{l+m=n}(-2\Delta_{21})_{l}\frac{a_{m}}{l!}$$

(13)

must be non negatives.

Although these conditions seem different from the proven positivity of the coefficients of the expansion of $g$ itself mentioned previously in the literature (See Appendix A of Hartman:2015 ), it can be seen that both are equivalent conditions. If we use the symmetry of the 4-point function under permutation $1\leftrightarrow 2$ and $3\leftrightarrow 4$, one can easily check that this implies the following identity:

$$(1-x)^{2\Delta_{21}}g^{21}_{21}(x)=g^{12}_{12}(x)$$

(14)

Using this equality one can see that $(1-x)^{2\Delta_{21}}g^{21}_{21}(x)$ is a CM function of $\rho$ for all values of $\Delta_{1}$ and $\Delta_{2}$ if and only if $g^{12}_{12}(x)$ is a CM function of $\rho$. And this last condition is equivalent to demanding that the coefficients “$a$” appearing in the expansion of $g$ in powers of $x$ also must be non-negatives, in agreement with Hartman:2015 .

If we choose as the parameter defining the sequences $X_{i}$ in the $\eta-$family the normalized distance $\eta_{i}=t_{i}/L$, one can see that the cross-ratio $x$ is a function of the sum variable $\eta=\eta_{i}+\eta_{j}$: $x=\frac{1}{(1+\eta_{i}+\eta_{j})^{2}}=\frac{1}{(1+\eta)^{2}}$. We can also check that this is the case for the other factors in (9) and it give us:

$$S_{4}(x_{1},x_{2},x_{3},x_{4})=L^{-2\Delta_{1}-2\Delta_{2}}\left(\frac{(1-x(\eta))^{2}}{x(\eta)}\right)^{\Delta_{21}}\eta^{-2\Delta_{21}}g^{21}_{21}(x(\eta))$$

(15)

that has to be a pd function of the variable $\eta$. Moreover, since this is bounded as $\eta\rightarrow\infty$, it implies that:

$$S_{4}(x_{1},x_{2},x_{3},x_{4})=L^{-2\Delta_{1}-2\Delta_{2}}\left(\frac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}g^{21}_{21}\left(\frac{1}{(\eta+1)^{2}}\right),$$

(16)

has to be completely monotonic (CM) as function of $\eta$. Note that (16) differs from (11) in the factors in front of $g$.

From the $\rho$ case in section 3.1 we see that $g^{21}_{21}(x)$ has a power expansion with positive coefficients of $x=\frac{1}{(1+\eta)^{2}}$. For any positive number $\alpha$, $\left(\frac{1}{(1+\eta)^{2}}\right)^{\alpha}$ is a CM function. Therefore as $g^{21}_{21}(\eta)$ is a combination of CM functions with positive coefficients. So when $\Delta_{21}=0$ the positivity conditions of the $\rho$ case imply that the conditions in the $\eta$ case are also fulfilled.

In the general case of $\Delta_{21}\neq 0$, we need to take into account the extra factor:

$$\left(\frac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}=\left(1+\sqrt{x(\eta)}\right)^{2\Delta_{21}}$$

that differs from the factor $(1-x)^{2\Delta_{21}}$ appearing in the positivity conditions (11) of $\rho$. So in order to study the CM character of (16), it is convenient to consider two separated case.

Case $\Delta_{2}>\Delta_{1}$: Let us recall the form of the function (16) which should be CM as a function of $\eta$ :

$$S_{4}(x_{1},x_{2},x_{3},x_{4})=\left(\frac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}L^{-2\Delta_{1}-2\Delta_{2}}g^{21}_{21}(x(\eta))$$

Using the definitions and theorems of section 2.1, one can see that $\left(\frac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}$ is a CM function. This can easily be done by showing that its logarithm (when $\Delta_{21}>0$) is a CM function and so it will also be its exponential.

Therefore, $S_{4}$ is a CM function of $\eta$, as it is a sum with positive coefficients in which each term is a product of CM functions, $\left(\tfrac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}\left(\frac{1}{(1+\eta)^{2}}\right)^{\alpha}$.

Case $\Delta_{2}<\Delta_{1}$: This case is more involved, since in this case $\left(\frac{\eta+2}{\eta+1}\right)^{2\Delta_{21}}$ is not a complete monotonic function. So, the CM character of $S_{4}$ is not guaranteed by the positivity of coefficients in $g$, meaning that in this case we have extra conditions. The CM character of $S_{4}(\eta)$ means that:

$$(-1)^{n}S_{4}^{(n)}(\eta)\geq{0},$$

or equivalently, that its Laplace anti-transform is non-negative.

In this subsection, we want to see how the previous considered positivity conditions are fulfilled in the conformal block itself. We will focus in the particular case of 2d CFT, considering all the Virasoro descendants. In this 2 dimensional case the check can be done in an elegant way and we use it to illustrate the idea, but this positivity checks can be done, at least numerically, for conformal blocks in general dimensions.

Now, we use the conformal block decomposition of $g$:

$$g_{34}^{21}(x,\bar{x})=\sum_{p}C_{34}^{p}C_{12}^{p}\mathcal{F}_{34}^{21}(p|x)\bar{\mathcal{F}}_{34}^{21}(p|\bar{x}),$$

(17)

where

		$\displaystyle\mathcal{F}_{34}^{21}(p|x)=x^{h_{p}}\sum_{k=0}^{+\infty}F^{(p)}_{k}x^{k}$		(18)
		$\displaystyle\bar{\mathcal{F}}_{34}^{21}(p|\bar{x})=\bar{x}^{\bar{h}_{p}}\sum_{k=0}^{+\infty}\bar{F}^{(p)}_{k}\bar{x}^{k},$

where $x$ and $\bar{x}$ are the holomorphic and antiholomorphic cross-ratios, that are equal when the points are aligned and ordered, and $\Delta_{p}=h_{p}+\bar{h}_{p}$, $\ell_{p}=h_{p}-\bar{h}_{p}$ are the conformal dimension and spin of the exchange fields.

RP inequalities applies in the case in which $1=4$ and $2=3$. In this case, we have:

$$g_{21}^{21}(x,\bar{x})=\sum_{p}(C_{12}^{p})^{2}\mathcal{F}_{21}^{21}(p|x)\bar{\mathcal{F}}_{21}^{21}(p|\bar{x}),$$

(19)

As we have mentioned in the introduction, we known from RychkovRP that RP is automatically fulfilled in the 4-point functions, without restricting the OPE coefficients or the field dimensions (besides unitary bound and reality of the OPE coefficient). Then, from the simple fact that the $C$’s appears as $C^{2}$ and they are real, from the positivity of the coefficients of $g$ we conclude that the conformal blocks should also have an expansion with positive coefficients, because if not, there will be further restrictions on the CFT data³³3Actually, what appears is $C_{21}^{p}C_{12}^{p}=(-1)^{\ell_{p}}C_{12}^{2}$, where $\ell_{p}$ is the spin of the exchange field. The minus sign for odd spin cancels with other minus from the conformal blocks.. This can be checked explicitly using the exact form of the conformal blocks.

The strategy can be understood in the following pedagogical example. Let us consider the Ising model in $d=2$, which is a minimal model corresponding to $c=\frac{1}{2}$ with three spinless primary fields of total conformal dimension $\Delta=h+\bar{h}=0,1/8,1$ ($h=\bar{h}$). As usual, let us denote these fields with $1,\sigma,\epsilon$ respectively. The OPE coefficients are partially determined by the fusion rules, which says in particular that:

$$[\sigma\sigma]=[1]+[\epsilon]$$

It means that the OPE coefficients $C_{\sigma\sigma}^{[\sigma]}=0$. $C_{\sigma\sigma}^{[1]}$ can be normalized to 1 while $C_{\sigma\sigma}^{[\epsilon]}$ ($C$ from now on) is not fixed by the fusion rules alone. However, the value of $C$ is completely fixed by CS, which impose that:

	$\displaystyle G^{\sigma\sigma}_{\sigma\sigma}(x)$	$\displaystyle=$	$\displaystyle G^{\sigma\sigma}_{\sigma\sigma}(1-x)\ \text{}\rightarrow$
	$\displaystyle x^{-2\Delta_{\sigma}}(F^{\sigma\sigma}_{\sigma\sigma}[1](x)+C^{2}F^{\sigma\sigma}_{\sigma\sigma}[\epsilon](x))$	$\displaystyle=$	$\displaystyle(1-x)^{-2\Delta_{\sigma}}(F^{\sigma\sigma}_{\sigma\sigma}[1](1-x)+C^{2}F^{\sigma\sigma}_{\sigma\sigma}[\epsilon](1-x))$

The $F^{\sigma\sigma}_{\sigma\sigma}[1]$ and $F^{\sigma\sigma}_{\sigma\sigma}[\epsilon]$ are the conformal blocks corresponding to the exchange fields $1$ and $\epsilon$, which are completely fixed by general properties of the CFT:

	$\displaystyle F[1](x)=\frac{1}{(1-x)^{\frac{1}{4}}}(1+\sqrt{1-x})$		(25)
	$\displaystyle F[\epsilon](x)=\frac{1}{(1-x)^{\frac{1}{4}}}(1-\sqrt{1-x})$

where we have suppressed the index $\sigma$.

Witting down the rhs of te CS equation:

$$rhs=\frac{1}{(1-x)^{\frac{1}{4}}}(1+C^{2}+(1-C^{2})\sqrt{x})$$

(26)

one can see that, in order to get a CM function of $\rho$ in the rhs, the coefficient of $\sqrt{x}$ must be not negative: $C^{2}\leq{1}$.

This example illustrates the main idea of this paper: the lhs is automatically CM, since the conformal blocks itself satisfies this conditions and they are combined with non-negative coefficients. Then the rhs should be a CM function of $\rho$ as well. But this is not guaranteed by the form of the conformal blocks. In fact, the first term is a CM function of $\rho$ but the second term is not. This imposes an upper bound to the non-CM term, which in this case turns to be $C^{2}\leq{1}$. Let us formulate the procedure in the more complicated case where we have an infinite number of OPE coefficients.

Let us write again the crossing symmetry equation for four identical scalar fields of conformal dimension $\Delta_{0}$, but now isolating the contribution of the identity at both sides:

$$\frac{1}{x^{2\Delta_{0}}}\left[1+\sum_{\Delta,\ell}C^{2}_{\Delta,\ell}\ F^{\Delta,\ell}(x)\right]=\frac{1}{(1-x)^{2\Delta_{0}}}\left[1+\sum_{\Delta,\ell}C^{2}_{\Delta,\ell}\ F^{\Delta,\ell}(1-x)\right]$$

(27)

where the sum is over all conformal dimension $\Delta$ (except the identity $\Delta=0$) and spin $\ell$ of the exchange fields, and we have simplify the notation in the $C$’s suppressing the external field label $\Delta_{0}$.

We can see that in the rhs we have the conformal block corresponding to the identity, i.e., the term $\frac{1}{(1-x)^{2\Delta_{0}}}$. This term is already a CM function and therefore does not help in our strategy of bounding the OPE coefficients demanding the rhs to be CM. As adding a pd function would lead to less constrains, it is better not to have pd functions in the rhs. So, to get a CM lhs, and a rhs without the contribution of the identity, our strategy will be to rewrite the equation in the following way:

$$1+\frac{1}{1-(\frac{x}{1-x})^{2\Delta_{0}}}\sum_{\Delta,\ell}C^{2}_{\Delta,\ell}\ F^{\Delta,\ell}(x)=\frac{x^{2\Delta_{0}}}{(1-x)^{2\Delta_{0}}-x^{2\Delta_{0}}}\sum_{\Delta,\ell}C^{2}_{\Delta,\ell}\ F^{\Delta,\ell}(1-x)$$

(28)