\(p\)-adic analysis and mathematical physics.

*(English)*Zbl 0812.46076
Series on Soviet and East European Mathematics. 10. Singapore: World Scientific. xviii, 319 p. (1994).

The book offers an introduction to a new branch of mathematical physics. To explain the title recall that a valuation on a field \(K\) is a map \(|\, |: K\to [0,\infty)\) satisfying \(| x|= 0\) iff \(x= 0\), \(| xy|= | x| | y|\), \(| x+ y|\leq | x|+ | y|\) \((x,y\in K)\). It induces naturally a metric \((x,y)\to | x- y|\) which opens the way to analysis in \(K\). On the rational number field \(\mathbb Q\) there exist by the Ostrowski Theorem essentially only the ordinary absolute value \(| \,|\) and, for each prime \(p\), the \(p\)-adic valuation \(|\, |_ p\) determined by \(| n|_ p= 1\) if \(n\in \mathbb Z\) is not divisible by \(p\), \(| p|_ p= p^{-1}\).

The \(p\)-adically valued field \(\mathbb Q\) and its completion \(\mathbb Q_ p\), the field of the \(p\)-adic numbers, are ‘non-archimedean’ in the sense that the sequence \(1,2,3,\dots\) is bounded. This is what makes ‘\(p\)-adic analysis’ so different from analysis over the real or complex numbers. In the past few decades \(p\)-adic analysis came to prosperity and attracted mathematicians from various disciplines. In 1984 a paper of the first two authors [Theor. Math. Phys. 59, 317–335 (1984); translation from Teor. Mat. Fiz. 59, 3–27 (1984; Zbl 0552.46023)] has ushered in the entrance of \(p\)-adic analysis into mathematical physics. With this book the mathematical community, for the first time, is offered easy access.

The authors are motivated by two basic considerations. First, it is mentioned that in quantum gravity and string theory there is a positive lower bound to all possible measuring of distances, the so-called Planck length, about \(10^{-33}\) cm. This suggests that one should drop the archimedean axiom of the real numbers which states essentially that two scales can be compared for small lengths. Secondly, it is pointed out that results of practical action are being expressed in rational numbers only. This, together with the above Ostrowski Theorem leads more or less naturally to the idea that \(p\)-adic numbers could become a tool for physical models at the Planck scale.

The first two chapters deal with \(p\)-adic analysis as far as it may be useful to physics. Here some wavering between two points of view can be observed: should one use functions \(\mathbb Q_ p\to \mathbb Q_ p\) (then integration is a problem) or functions \(\mathbb Q_ p\to \mathbb C\) (now one can use Haar measure to integrate, but differentiation fails)? As it turns out most attention is focused on complex-valued functions on a \(p\)-adic domain. [In this context one cannot disregard the book of A. Khrennikov, \(p\)-adic valued distributions in mathematical systems. Dordrecht: Kluwer (1994; Zbl 0833.46061), treating the same subject but, alternatively, has the emphasis on \(\mathbb Q_ p\)-valued functions.]

Chapter 1 contains basics, Pontryagin duality of additive and multiplicative (complex valued) characters \(\chi_ p\) on \(\mathbb Q_ p\) and its quadratic extensions \(\mathbb Q_ p(\sqrt\varepsilon)\) (there does not seem to be a need for the \(p\)-adic complex number field \(\mathbb C_ p\)). Formulas for Gaussian integrals of the type \(\int_{\mathbb Q_ p} \chi_ p(ax^ 2+ bx)\,dx\) are obtained. Distribution theory à la Bruhat, Fourier theory, convolution of generalized functions on \(\mathbb Q^ n_ p\) are treated. The role of the test function space is taken over by the set \(\mathcal D\) of all locally constant functions with compact support.

Chapter 2 offers a substitute for differentiation of functions \(\mathbb Q_ p\to \mathbb C_ p\) in the form of pseudodifferential operators \(D^ \alpha\) \((\alpha>0)\) and primitivation operators \(D^ \alpha\) \((\alpha< 0)\) working on distributions \(\psi\). If \(\psi\in {\mathcal D}\) then \(D\) has the form \((D\psi)(x)= \int_{\mathbb Q_ p} |\xi|_ p \widetilde\psi(\xi) \chi_ p(-\xi x) \,d\xi\). In both chapters numerous examples and concrete computations are being carried out.

The last chapter heavily uses the machinery of the two foregoing ones and deals with \(p\)-adic quantum mechanics and its spectral theory, Weyl systems, \(p\)-adic strings, quantum groups and \(q\)-analysis, stochastic processes over the \(p\)-adic numbers. It offers a daring, mathematical \(p\)-adic translation of existing formalisms in classical mechanics and quantum theory. The question as to whether the resulting theory is ‘valid’ or ‘useful’ remains open and, of course, the ultimate answer is reserved to physicists.

The \(p\)-adically valued field \(\mathbb Q\) and its completion \(\mathbb Q_ p\), the field of the \(p\)-adic numbers, are ‘non-archimedean’ in the sense that the sequence \(1,2,3,\dots\) is bounded. This is what makes ‘\(p\)-adic analysis’ so different from analysis over the real or complex numbers. In the past few decades \(p\)-adic analysis came to prosperity and attracted mathematicians from various disciplines. In 1984 a paper of the first two authors [Theor. Math. Phys. 59, 317–335 (1984); translation from Teor. Mat. Fiz. 59, 3–27 (1984; Zbl 0552.46023)] has ushered in the entrance of \(p\)-adic analysis into mathematical physics. With this book the mathematical community, for the first time, is offered easy access.

The authors are motivated by two basic considerations. First, it is mentioned that in quantum gravity and string theory there is a positive lower bound to all possible measuring of distances, the so-called Planck length, about \(10^{-33}\) cm. This suggests that one should drop the archimedean axiom of the real numbers which states essentially that two scales can be compared for small lengths. Secondly, it is pointed out that results of practical action are being expressed in rational numbers only. This, together with the above Ostrowski Theorem leads more or less naturally to the idea that \(p\)-adic numbers could become a tool for physical models at the Planck scale.

The first two chapters deal with \(p\)-adic analysis as far as it may be useful to physics. Here some wavering between two points of view can be observed: should one use functions \(\mathbb Q_ p\to \mathbb Q_ p\) (then integration is a problem) or functions \(\mathbb Q_ p\to \mathbb C\) (now one can use Haar measure to integrate, but differentiation fails)? As it turns out most attention is focused on complex-valued functions on a \(p\)-adic domain. [In this context one cannot disregard the book of A. Khrennikov, \(p\)-adic valued distributions in mathematical systems. Dordrecht: Kluwer (1994; Zbl 0833.46061), treating the same subject but, alternatively, has the emphasis on \(\mathbb Q_ p\)-valued functions.]

Chapter 1 contains basics, Pontryagin duality of additive and multiplicative (complex valued) characters \(\chi_ p\) on \(\mathbb Q_ p\) and its quadratic extensions \(\mathbb Q_ p(\sqrt\varepsilon)\) (there does not seem to be a need for the \(p\)-adic complex number field \(\mathbb C_ p\)). Formulas for Gaussian integrals of the type \(\int_{\mathbb Q_ p} \chi_ p(ax^ 2+ bx)\,dx\) are obtained. Distribution theory à la Bruhat, Fourier theory, convolution of generalized functions on \(\mathbb Q^ n_ p\) are treated. The role of the test function space is taken over by the set \(\mathcal D\) of all locally constant functions with compact support.

Chapter 2 offers a substitute for differentiation of functions \(\mathbb Q_ p\to \mathbb C_ p\) in the form of pseudodifferential operators \(D^ \alpha\) \((\alpha>0)\) and primitivation operators \(D^ \alpha\) \((\alpha< 0)\) working on distributions \(\psi\). If \(\psi\in {\mathcal D}\) then \(D\) has the form \((D\psi)(x)= \int_{\mathbb Q_ p} |\xi|_ p \widetilde\psi(\xi) \chi_ p(-\xi x) \,d\xi\). In both chapters numerous examples and concrete computations are being carried out.

The last chapter heavily uses the machinery of the two foregoing ones and deals with \(p\)-adic quantum mechanics and its spectral theory, Weyl systems, \(p\)-adic strings, quantum groups and \(q\)-analysis, stochastic processes over the \(p\)-adic numbers. It offers a daring, mathematical \(p\)-adic translation of existing formalisms in classical mechanics and quantum theory. The question as to whether the resulting theory is ‘valid’ or ‘useful’ remains open and, of course, the ultimate answer is reserved to physicists.

Reviewer: Wim Schikhof (Nijmegen)

##### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81Q65 | Alternative quantum mechanics (including hidden variables, etc.) |

26E30 | Non-Archimedean analysis |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

30G06 | Non-Archimedean function theory |