The Divergence of Van Hove’s Model and its Consequences

Abstract

We study a regularized version of Van Hove’s 1952 model, in which a quantum field interacts linearly with sources of finite width lying at fixed positions. We show that the central result of Van Hove’s 1952 paper on the foundations of Quantum Field Theory, the orthogonality between the spaces of state vectors which correspond to different values of the parameters of the theory, disappears when a well-defined model is considered. We comment on the implications of our results for the contemporary relevance of Van Hove’s article.

Appendix

1.1 Results About the Infinite Sums

In this appendix we collect results about the infinite sums in (17) and (18). As we see below, the character of convergence of all these sums can be traced back to that of the infinite sum

$$\begin{aligned} \frac{1}{V} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \, \frac{1}{\omega _{k}^{2}} \, \big |\tilde{\rho }({\mathbf {k}})\big |^{2} \quad , \end{aligned}$$

(48)

where $\tilde{\rho }({\mathbf {k}})$ is the Fourier coefficient of a function $\rho$ periodic on the cubic lattice and compactly supported in its primitive cell ${\mathscr {P}}\,$.

A word about the notation. In what follows, a triple $\alpha = (\alpha _{_{1}}, \alpha _{_{2}}, \alpha _{_{3}})$ of non-negative integers is called a tri-index, and the space of tri-indices is indicated with $I_{^{+}}^{3}\,$. We indicate

$$\begin{aligned} {\mathbf {x}}^{\alpha }&= x_{_{1}}^{\alpha _{_{1}}} \, x_{_{2}}^{\alpha _{_{2}}} \, x_{_{3}}^{\alpha _{_{3}}} \quad ,&\partial ^{\beta }&= \partial _{_{1}}^{\beta _{_{1}}} \, \partial _{_{2}}^{\beta _{_{2}}} \, \partial _{_{3}}^{\beta _{_{3}}} \quad . \end{aligned}$$

(49)

We moreover indicate with $C_{c}^{\infty }$ the set of smooth functions with compact support, and ${\mathscr {S}}$ indicates the Schwartz space of rapidly decreasing functions which by definition satisfy

$$\begin{aligned} \sup _{{\mathbf {x}}\in \mathbb {R}^{3}} \big |{\mathbf {x}}^{\alpha } \, \partial ^{\beta } f\big | < \infty \quad , \qquad \forall \, \alpha , \beta \in I_{^{+}}^{3} \quad . \end{aligned}$$

(50)

1.2 Convergence

To understand the behavior of $\tilde{\rho }({\mathbf {k}})$ when $k \rightarrow \infty \,$, it is useful to introduce an auxiliary function r closely related to $\rho \,$. While the latter is periodic on the lattice, and it can be thought as being defined on the primitive cell ${\mathscr {P}}$ and extended by periodicity, let us consider a non-periodic function $r : \mathbb {R}^{3} \rightarrow \mathbb {R}$ which coincides with $\rho$ only on the primitive cell, that is

$$\begin{aligned} r({\mathbf {x}}) = {\left\{ \begin{array}{ll} \, \rho ({\mathbf {x}})&{} \text {if} \,\, {\mathbf {x}}\in {\mathscr {P}}\quad , \\ \, 0&{} \text {elsewhere} \quad . \end{array}\right. } \end{aligned}$$

(51)

Clearly, r belongs to $C_{c}^{\infty }(\mathbb {R}^{3})$ as a consequence of $\rho$ belonging to $C_{c}^{\infty } ({\mathscr {P}})\,$. Defining the Fourier transform of r as

$$\begin{aligned} \tilde{r}({\mathbf {k}}) = \int \limits _{\mathbb {R}^{3}} \! r({\mathbf {x}}\, ) \, e^{-i {\mathbf {k}}\cdot {\mathbf {x}}} \, d^{3}x \quad , \end{aligned}$$

(52)

the comparison of (9) with (52) reveals that, by the way we defined r, we have

$$\begin{aligned} \tilde{r}({\mathbf {k}}) = \tilde{\rho }({\mathbf {k}}) \quad , \qquad \forall \, {\mathbf {k}}\in {\mathcal {R}}\quad . \end{aligned}$$

(53)

In other words, the Fourier transform of r evaluated at a vector ${\mathbf {k}}$ of the reciprocal lattice coincides with the Fourier coefficient of $\rho$ evaluated at the same ${\mathbf {k}}\,$. With this in mind, it is possible to work with $\tilde{r}$ instead of with $\tilde{\rho }$ whenever convenient.

Recall now that the Fourier transform, as an operator, is a bijection of ${\mathscr {S}}(\mathbb {R}^{3})$ onto ${\mathscr {S}}(\mathbb {R}^{3})$ [19]. Since $C_{c}^{\infty }(\mathbb {R}^{3}) \subset {\mathscr {S}}(\mathbb {R}^{3})$ it follows that $r \in {\mathscr {S}}(\mathbb {R}^{3})\,$, and therefore $\tilde{r} \in {\mathscr {S}}(\mathbb {R}^{3})$ as well. By the property (50), the fact that $\tilde{r}$ is rapidly decreasing implies that for every $\alpha \in I_{^{+}}^{3}$ we have

$$\begin{aligned} \sup _{{\mathbf {k}}\in \mathbb {R}^{3}} \, \bigg |\, {\mathbf {k}}^{\alpha } \, \frac{1}{\omega _{k}^{2}} \, \big |\tilde{r}({\mathbf {k}})\big |^{2}\bigg | < \infty \quad , \end{aligned}$$

(54)

and taking into account (53) we get

$$\begin{aligned} \sup _{{\mathbf {k}}\in {\mathcal {R}}} \, \bigg |\, {\mathbf {k}}^{\alpha } \, \frac{1}{\omega _{k}^{2}} \, \big |\tilde{\rho }({\mathbf {k}})\big |^{2}\bigg | < \infty \quad , \end{aligned}$$

(55)

again for every $\alpha \in I_{^{+}}^{3}$. In particular, $\big |\tilde{\rho }({\mathbf {k}})\big |^{2}/\omega _{k}^{2}$ goes to zero more rapidly than any non-negative power of $1/k\,$. Since the number of points of the reciprocal lattice contained in a sphere of radius k diverge as a power of k when $k \rightarrow \infty$, it follows that the infinite sum (48) converges absolutely.

This result readily applies to the convergence of the infinite sums in (18), since both $\rho _{_{1}\!}$ and $\rho _{_{2}\!}$ are compactly supported. Regarding (17), note that

$$\begin{aligned}&\bigg | \, \mathfrak {R}\Big [ \, \tilde{\rho }_{_{1}}\!({\mathbf {k}}) \, \tilde{\rho }^{*}_{_{2}}({\mathbf {k}}) \, \Big ] \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {y}}_{_{\!1\!}} - {\mathbf {y}}_{_{\!2\!}} \big ) \Big ] + \mathfrak {I}\Big [ \, \tilde{\rho }_{_{1}}\!({\mathbf {k}}) \, \tilde{\rho }^{*}_{_{2}}({\mathbf {k}}) \, \Big ] \, \sin \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {y}}_{_{\!1\!}} - {\mathbf {y}}_{_{\!2\!}} \big ) \Big ]\bigg | \le \nonumber \\&\quad \le \bigg | \, \mathfrak {R}\Big [ \, \tilde{\rho }_{_{1}}\!({\mathbf {k}}) \, \tilde{\rho }^{*}_{_{2}}({\mathbf {k}}) \, \Big ]\bigg | + \bigg |\mathfrak {I}\Big [ \, \tilde{\rho }_{_{1}}\!({\mathbf {k}}) \, \tilde{\rho }^{*}_{_{2}}({\mathbf {k}}) \, \Big ]\bigg | \le 2 \, \big |\tilde{\rho }_{_{1}}\!({\mathbf {k}})\big | \, \big |\tilde{\rho }_{_{2}}\!({\mathbf {k}})\big | \quad , \end{aligned}$$

(56)

and recall that the product of two rapidly decreasing functions is rapidly decreasing. It follows that the infinite sum in (17) converges absolutely.

1.3 Infinite Volume Limit

We now consider the infinite volume limit

$$\begin{aligned} \lim _{V \rightarrow \infty } \, \frac{1}{V} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \, \frac{1}{\omega _{k}^{2}} \,\, \big |\tilde{\rho }({\mathbf {k}})\big |^{2} \quad . \end{aligned}$$

(57)

1.4 Existence

Note first of all that

$$\begin{aligned} \frac{1}{V} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \, \frac{1}{\omega _{k}^{2}} \,\, \big |\tilde{\rho }({\mathbf {k}})\big |^{2} = \frac{1}{(2 \pi )^{3}} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \bigg ( \frac{2 \pi }{L} \bigg )^{\!\! 3} \frac{1}{\omega _{k}^{2}} \,\, \big |\tilde{r}({\mathbf {k}})\big |^{2} \quad , \end{aligned}$$

(58)

and that the infinite sum on the right hand side is a Riemann sum of the integral

$$\begin{aligned} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}({\mathbf {k}})\big |^{2} \, d^{3}k \quad . \end{aligned}$$

(59)

Specifically, the cells of the reciprocal lattice are the 3-intervals of the Riemann sum.

The convergence of the integral (59) is easy to establish using the results of the previous section. In fact, writing the integral as the limit for $K \rightarrow \infty$ of the related integral on the sphere ${\mathcal {B}}(K)$ of radius K centered at the origin, we have

$$\begin{aligned} \lim _{K \rightarrow \infty } \int \limits _{{\mathcal {B}}(K)} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}({\mathbf {k}})\big |^{2} \, d^{3}k \le \lim _{K \rightarrow \infty } \int \limits _{0}^{K} \frac{4 \pi k^{2}}{\omega _{k}^{2}} \max _{\Vert {\mathbf {z}}\Vert = k} \big |\tilde{r}({\mathbf {z}})\big |^{2} dk \quad , \end{aligned}$$

and since $\tilde{r} \in {\mathscr {S}}(\mathbb {R}^{3})$ the limit comparison test with the function $k^{-2}$ permits to positively conclude about the convergence. The expression (57) is therefore (proportional to) the limit of Riemann sums of a convergent integral, when the volume of the 3-intervals of the Riemann sums tend to zero. It follows that the infinite volume limit (57) exists, and

$$\begin{aligned} \lim _{V \rightarrow \infty } \frac{1}{V} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \, \frac{1}{\omega _{k}^{2}} \,\, \big |\tilde{\rho }({\mathbf {k}})\big |^{2} = \frac{1}{(2 \pi )^{3}} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}({\mathbf {k}})\big |^{2} \, d^{3}k \quad . \end{aligned}$$

(60)

1.5 Potential-Mediated Expression

The expression on the right hand side of (60) can be recast in a perhaps more intuitive form. Note that, since $r({\mathbf {x}}) \, r({\mathbf {z}}) \in C^{\infty }_{c}(\mathbb {R}^{6})\,$, using Fubini’s theorem and the definition (52) we have

$$\begin{aligned} \big |\tilde{r}({\mathbf {k}})\big |^{2} = \tilde{r}^{*}({\mathbf {k}}) \, \tilde{r}({\mathbf {k}}) = \iint \limits _{\mathbb {R}^{6}} \! r({\mathbf {x}}) \, r({\mathbf {z}}) \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {x}}- {\mathbf {z}}\big ) \Big ] \, d^{3}x \, d^{3}z \quad , \end{aligned}$$

(61)

so

$$\begin{aligned} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}({\mathbf {k}})\big |^{2} \, d^{3}k = \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \Bigg \{ \iint \limits _{\mathbb {R}^{6}} \! r({\mathbf {x}}) \, r({\mathbf {z}}) \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {x}}- {\mathbf {z}}\big ) \Big ] \, d^{3}x \, d^{3}z \Bigg \} \, d^{3}k \quad . \end{aligned}$$

(62)

It would be attractive to express (62) as the integral in $d^{3}x \, d^{3}z$ of the product of the charge distributions mediated by a potential. As is well-known this is indeed possible, and it results into the appearance of the Yukawa potential mediating the interaction

$$\begin{aligned} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}({\mathbf {k}})\big |^{2} \, d^{3}k = 2 \pi ^{2} \!\! \iint \limits _{\mathbb {R}^{6}} r({\mathbf {x}}) \, r ({\mathbf {z}}) \, \frac{e^{- m \Vert {\mathbf {x}}- {\mathbf {z}}\Vert }}{\Vert {\mathbf {x}}- {\mathbf {z}}\Vert } \, d^{3} x \, d^{3}z \quad . \end{aligned}$$

(63)

These relations allow us to express the numbers B and C in the forms (21) and (22). Regarding C, consider the expression (18) and for each function $\rho _{i}$ introduce the associate auxiliary function $r_{i}$ defined as in Sect A.1. The relation (60) then implies that the infinite volume limit of C takes the form

$$\begin{aligned} C \xrightarrow [V \rightarrow \infty ]{} - \frac{1}{2} \, \sum _{i = 1}^{2} \,\, \frac{g_{i}^{2}}{(2 \pi )^{3}} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \, \big |\tilde{r}_{i}({\mathbf {k}})\big |^{2} \, d^{3}k \quad , \end{aligned}$$

(64)

and using (63) we arrive at

$$\begin{aligned} C \xrightarrow [V \rightarrow \infty ]{} - \frac{1}{2} \, \sum _{i = 1}^{2} \,\, g_{i}^{2} \iint \limits _{\mathbb {R}^{6}} r_{i}({\mathbf {x}}) \, r_{i}({\mathbf {z}}) \, \frac{1}{4 \pi } \, \frac{e^{- m \Vert {\mathbf {x}}- {\mathbf {z}}\Vert }}{\Vert {\mathbf {x}}- {\mathbf {z}}\Vert } \, d^{3} x \, d^{3}z \quad , \end{aligned}$$

(65)

which is (22).

Regarding B, note that (17) can be equivalently written as

$$\begin{aligned} B = - \frac{g_{_{1}} g_{_{2}}}{V} \, \sum _{{\mathbf {k}}\in {\mathcal {R}}} \, \frac{1}{\omega _{k}^{2}} \, \mathfrak {R}\bigg [ \, \tilde{\rho }_{_{1}}\!({\mathbf {k}}) \, \tilde{\rho }^{*}_{_{2}}({\mathbf {k}}) \, e^{-i {\mathbf {k}}\cdot \big ( {\mathbf {y}}_{_{\!1\!}} - {\mathbf {y}}_{_{\!2\!}} \big )} \bigg ] \quad , \end{aligned}$$

(66)

and that the same arguments of Sect A.2.1 can be used to infer that its infinite volume limit exists and takes the form

$$\begin{aligned} B \xrightarrow [V \rightarrow \infty ]{} - \frac{g_{_{1}} g_{_{2}}}{(2 \pi )^{3}} \, \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \, \mathfrak {R}\bigg [ \, \tilde{r}_{_{\!1}}\!\!({\mathbf {k}}) \, \tilde{r}^{*}_{_{\!2}}({\mathbf {k}}) \, e^{-i {\mathbf {k}}\cdot \big ( {\mathbf {y}}_{_{\!1\!}} - {\mathbf {y}}_{_{\!2\!}} \big )} \bigg ] \, d^{3}k \quad . \end{aligned}$$

(67)

Furthermore, it is straightforward to verify that

$$\begin{aligned}&\mathfrak {R}\bigg [ \, \tilde{r}_{_{\!1}}\!\!({\mathbf {k}}) \, \tilde{r}^{*}_{_{\!2}}({\mathbf {k}}) \, e^{-i {\mathbf {k}}\cdot \big ( {\mathbf {y}}_{_{\!1\!}} - {\mathbf {y}}_{_{\!2\!}} \big )} \bigg ] \nonumber \\&\quad = \iint \limits _{\mathbb {R}^{6}} r_{_{\!1}} ({\mathbf {x}}- {\mathbf {y}}_{_{\!1\!}}) \, r_{_{\!2}} ({\mathbf {z}}- {\mathbf {y}}_{_{\!2\!}}) \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {x}}- {\mathbf {z}}\big ) \Big ] \, d^{3} x \, d^{3} z \quad , \end{aligned}$$

(68)

so the integral in (67) can be written as

$$\begin{aligned} \int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \,\, \Bigg \{ \iint \limits _{\mathbb {R}^{6}} \! r_{_{\!1}} ({\mathbf {x}}- {\mathbf {y}}_{_{\!1\!}}) \, r_{_{\!2}} ({\mathbf {z}}- {\mathbf {y}}_{_{\!2\!}}) \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {x}}- {\mathbf {z}}\big ) \Big ] \, d^{3}x \, d^{3}z \Bigg \} \, d^{3}k \quad . \end{aligned}$$

(69)

Analogously to the passage from (62) to (63), we have

$$\begin{aligned}&\int \limits _{\mathbb {R}^{3}} \frac{1}{\omega _{k}^{2}} \,\, \Bigg \{ \iint \limits _{\mathbb {R}^{6}} \! r_{_{\!1}} ({\mathbf {x}}- {\mathbf {y}}_{_{\!1\!}}) \, r_{_{\!2}} ({\mathbf {z}}- {\mathbf {y}}_{_{\!2\!}}) \, \cos \Big [ {\mathbf {k}}\cdot \big ( {\mathbf {x}}- {\mathbf {z}}\big ) \Big ] \, d^{3}x \, d^{3}z \Bigg \} \, d^{3}k \nonumber \\&\quad = 2 \pi ^{2} \!\! \iint \limits _{\mathbb {R}^{6}} r_{_{\!1}} ({\mathbf {x}}- {\mathbf {y}}_{_{\!1\!}}) \, r_{_{\!2}} ({\mathbf {z}}- {\mathbf {y}}_{_{\!2\!}}) \, \frac{e^{- m \Vert {\mathbf {x}}- {\mathbf {z}}\Vert }}{\Vert {\mathbf {x}}- {\mathbf {z}}\Vert } \, d^{3} x \, d^{3}z \quad , \end{aligned}$$

(70)

so we get

$$\begin{aligned} B \xrightarrow [V \rightarrow \infty ]{} - g_{_{1}} g_{_{2}} \iint \limits _{\mathbb {R}^{6}} r_{_{\!1}} ({\mathbf {x}}- {\mathbf {y}}_{_{\!1\!}}) \, r_{_{\!2}} ({\mathbf {z}}- {\mathbf {y}}_{_{\!2\!}}) \, \frac{1}{4 \pi } \, \frac{e^{- m \Vert {\mathbf {x}}- {\mathbf {z}}\Vert }}{\Vert {\mathbf {x}}- {\mathbf {z}}\Vert } \, d^{3} x \, d^{3}z \quad , \end{aligned}$$

(71)

which is (21).