Geometry of PDEs and mechanics

IQEOMETRYI

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1 MECHANICS!

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IQEOMETRY I

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PDEs PD' Es

| MECHANICS I Asostino Prastaro Diparttmento di Metodi e Modelli Matetnatici per le Scienze Applicate Universita degli Studi di Roma 'La Sapienza"

I World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 91280f USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Pr£staro, Agostino. Geometry of PDEs and mechanics / Agostino Prastaro. 760 p.; 22.5 cm. Includes bibliographical references and index ISBN 9810225202 1. Mechanics-Mathematics. 2. Geometry, Algebraic. 3. Differential equations, Partial. 4. Mathematical physics. T Title

QC125.2.P73 1996 530.1'55353--dc20

95-47858 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923, USA.

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

To the memory of my father and mother and To my wife Paola and my sons Francesco and Alessandrc

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PREFACE 'Non Non inorgogliamoci

troppo del perfetto

rigore a cui

crediamo

di essere oggi capaci di ridurre tanta parte della matematict matematic matematica della mat e■ non iiuit, scartiamo auui LIU.ULUquel yue*che c/tenon iiuit ci t.tpare puin deltieitutto LULVU rigoroso, iigvi vsu, poiche domani

si troveranno

certo imperfezioniom 1nella nostra'a peperfezione

e da qualche geniale atto aito di intuizione,ne, che non ha ancora i crismi del rigore, si troveranno

risultatiitl

277impensabili imvensabil impensabili'

Francesco Severi rrancesco Severi sever

This book is an introduction to Mechanics revisited with a modern geometric language. It is crowded, in some sense, around courses held at the University and researches in Mathematics and Mathematical Physics, that I developed until now. My principal interest in research is the geometrization of Physics and the study of geometry of partial differential equations. In fact, Physics is essentially geometry, so the most suitable language to use, in order to describe physical phenomena, is geometric. On the other hand, all Mathematics has been reformulated with an algebraic-geometric language by obtaining great results. Any physical theory is characterized by a ruiLi. li one takes seriously this simple observation, we can consider that the true universe that characterizes a physical theory is the manifold that identifies the corresponding PDE. For example, the manifold that characterizes the General Relativity is not a 4-dimensional manifold M (space-time), but a 144-dimensional manifold, (Ein), that corresponds to the geometric structure (Ein) C JT>2(S%M) that characterizes Einstein equation (Ein) . So, from this point of view, it becomes very important to study the geometric structures (manifolds) that describe PDEs. This structure is more important than the manifold of solutions, as it contains more information than the latter. But, the manifold that characterizes a PDE identifies also its set of solutions. So all the usual informations about PDEs can be recovered by studying the geometry of PDEs. Thus, in this book we study the geometry of PDEs first. Boundary value problems, whether for regular solutions or singular ones are studied. We characterize also PDEs by means of some useful spectral sequences and study differential cobordisms in PDEs. These allow us to give a topological algebraic characterization whether of quantum solutions or classic ones. Then, the general geometric framework of PDEs, allows us to give a very elegant description of mechanics and field theory, and to open new prospects in these areas. Of course, the most suitable presentation of any physical theory is in a space-time framework. So, here, we start by introducing the Galilean space-time, and, in this panorama, we build the meaning of admissible motions and fields, by emphasizing the geometric role of dynamic equations. Quantizations of classical theories, have natural places in this picture. More precisely, we consider the vii

formal quantization of partial differential equations, first introduced in refs. [103,104], and further developed in refs.[105,106,108,109,111,112,113,114]. This is a procedure that allows us to associate to any PDE a measure space (quantum situs). This quantization becomes effective if on the quantum situs we recognize a (pre-)spectral measure (quantum spectral measure). Then it is proved that canonical quantization is a phenomenon governed by the geometry of PDEs. The geometrical structure of the quantum situs, O(^oo) , of the infinity prolongation of a PDE Ek , is explicitly given. A structure of C£°—manifold [56] is recognized on £l(Eoo). It is proved that formal quantization allows us to give a geometric description of quantum tunnel effects in PDEs. Applications to some important equations of the field theory are given. In the last chapter a theory of non-commutative manifolds (quantum manifolds) is proposed. Quantum manifolds are seen as locally convex manifolds where the model has the structure Am~*~n , where A is a non-commutative algebra that satisfies some particular axioms (quantum algebra). For such manifolds we also formulate a geometry and a formal theory of PDEs. Furthermore, as we characterize quantum manifolds by means of some geometrical objects defined on the underlying topological manifold, we are able to develop a quantum theory simulating on microlevel what the gravitational theory makes on macrolevel one. In fact, we formulate a quantum physical theory where the dynamical variables (quantum structure) are defined on quantum manifolds, and are solutions of suitable PDEs. A general theory of quantum tunnel effects on such equations is obtained on the ground of cobordism theory and Morse theory. In this framework a model of quantum supergravity is given where the quantum structure interprets at the classic limit curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on space-time. We emphasize that in this book we have discussed some new results that are not yet published. For example, completely new results that are relative to the classifications of singular solutions by means of cobordisms in PDEs, and the geometry of quantum PDEs, and its applications to quantum field theory and quantum supergravity. Roma, July 1995. Agostino Prastaro

viii

CONTENTS 1.- ALGEBRAIC GEOMETRY 1.1- Algebraic Complements 1.2- Affine Spaces 1.3 - Differential Manifolds 1.4 - Grassmann Manifolds 1.5 - Spectral Sequences 2.- DIFFERENTIAL EQUATIONS (PDEs) 2.1 - Geometry of Differential Equations 2.2 - Ordinary Differential Equations (ODEs) 2.3 - Characteristics of PDEs 2.4 - Affine PDEs and Green Functions 2.5 - Spectral Sequences in PDEs 2.6 - Tunnel Effects in PDEs 2.7 - Cobordism Groups in PDEs 3.- MECHANICS 3.1 - Structure of Galilean Space-Time 3.2 - One-Body Dynamics 3.3 - Important Formulas 3.4 - Fundamental Theorems of Dynamics 3.5 - Lagrangian Mechanics for Perfect Holonomic Systems 3.6 - Rigid Body Dynamics 4.- CONTINUUM MECHANICS 4.1 - Flow 4.2 - Stress Tensor and Moment of Stress Tensor 4.3 - Local Dynamic Equations 4.4 - Thermodynamics of Continuum Media 4.5 - Rheological Classification of Materials 4.6 - Rheoptics 4.7 - Multicomponent Continuum Systems 4.8 - Variational Field Theory 5.- QUANTUM FIELD THEORY 5.1 - Locally Convex Manifolds and Derivative Spaces 5.2 - Differential Geometry of Quantum Situs 5.3 - Mathematical Logic and Quantization of PDEs 5.4 - Formal and Dirac Quantizations of PDEs 5.5 - Canonical Quantization of PDEs

ix

2 7 9 18 31 68 94 108 129 200 225 241 307 331 347 353 369 381 393 397 400 409 420 479 487 502 533 544 555 561 619

6. - GEOMETRY OF QUANTUM PDEs 6.1 - Differential Geometry of Quantum Manifolds 6.2 - Cohomology of Quantum Manifolds 6.3 - Formal Theory of Quantum PDEs 6.4 - Cartan Spectral Sequences of Quantum PDEs 6.5 - Quantum Distributions and Singular Solutions of Quantum PDEs 6.6 - Tunnel Effects in Quantum PDEs 6.7 - Quantum PDEs Non-holonomic Connections 6.8 - Gauge Quantum PDEs 6.9 - Supergravity Quantum PDEs References : Index List of Symbols

x

626 663 671 681 694 701 706 715 719 725 733 749

1 - ALGEBRAIC GEOMETRY

1.1 - A L G E B R A I C C O M P L E M E N T S The purpose of this section is to recall some fundamental definitions and results of Al­ gebra that will be used in the next chapters. (For more details see also refs. [1,7,14,43, 49,53,79,80,86].) DEFINITION 1.1 - A matrix is a map: f : Jj x • • • x Ik -+ R ,

/ : (ti, ...,»*)-► / i i ^ . . . h £

R

where I\ = {1, • • •, n i } , • ■ •, J* = {1, • • •, rik). We will denote the set of matrices with k indexes M n i n 2 T l 3 ... njfc, where the lower and upper indexes are called covariant and controvariant respectively. EXAMPLE 1.1 - 1) k = 2, m = n, n 2 = m, h = I = {1, • • •, n}, J 2 = J = {1, • • •, m } , /:IxJ-*R

,

f:(ij)*-+fij,

where /»j-is the generic element of the matrix. In this case one can represent the matrix in the following way: /n |_ / n l

*••

hm

" * * Jnm J

This is an example of 2 x 2 matrix with two covariant indexes. 2) (fiJ) belongs to the matrix-space Mnm . The first index is covariant, the second one is controvariant. ■ PROPOSITION 1.1 - The set of matrices of some type is a vector space on R . EXAMPLE 1.2 - For example, in Mnrn we have: a) (/i + f2)ij = {fi)ij + (f2)ij (addition) b) (A/),-j = Xfij (scalar multiplication) The zero of Mnm is the matrix (0,-j), with 0,-j = 0 G R; the opposite of (fij) is (—fij) • In fact, for any index i,j one has: fij — fij = 0 . ■ DEFINITION 1.2 - V* = L(V\R). V* is the space of linear applications V -» R. We call V* the dual of V. REMARK 1.1 - One has the following non-canonical isomorphisms. (V^Mn-

V* ^ M n ; L(V; W) ^ M™; L(R; W) ^ W ^ M m m

L ( y ; R ) = V * ^ M n ; X ^ , ^ ; W) = Mnin2 i Z(y;,V 2 *;W) = M » ^

ro

L{V?, V2] W) = M

; ^ , 7 2 ; ^ ) = ^ ^ ;

lL(y1,...,Vp;^)EEMni...npm.

] ni

„2

m

X(F1*,F2;^)EEM^n2m I

J 2

Here L(V; W) denotes the space of all linear applications V —► W. We will also use the following notation HomK(V;W) = L(V\ W). Furthermore, L(VU • • •, Vp\ W) denotes the space of p-linear applications V\ x • • • x Vp —» W . DEFINITION 1.3 - We set GL(n; R ) = {a G M n n | det a ^ 0} . We caii GL(n\ R) the general linear n-dimensional real group. THEOREM 1.1 - The matrix that relates two different bases, in an n-dimensional vector space, belongs to GL(n] R ) . DEFINITION 1.4 - The tensor product between vector spaces V and W is V(g)W=

/I,

where < V x W > denotes the space of all linear combinations of couples (v, w), with v G V,w G W. I is the subspace in generated by means of vectors of the types: ( (v1+v2,w) — (vi,w) — (v2,w) \ I (\v,w) — \(v,w) < (v,w1+w2)-(v,w1)-(v,w2) ( (v,\w)

— \(v,w)

I > J

Let us denote by u (g) v the equivalence class identified by (u,v) E< V x W > . We say that u ®v is the tensor product ofu with v . THEOREM 1.2- 1) One has: dim V (g) W = nm , if dimV = n, dim W = m . 2) One has the following canonical isomorphisms: L(T/w)^v*(g)W;

L(y1,y2;W)^y1*(g)y2*(g)w.

DEFINITION 1.5 - A tensor of the t y p e (r,s) on V,&mV = n , is a vector of

r;(v)Ey0...r...0v0r0"v®r. WesetT 0 °(V) = R . PROPOSITION 1.2 - For any basis t{ in V one has the following linear representation: t = th'"irjx...jseh ®eir®0h ®---®0ja where (t u "" , r j 1 .-. J s ) G Mn'"r'nn...a...n. {^"}i<j< n is the dual basis of ti defined by the condition: # J (e;) = 8\ , where (Sj) the identity matrix. One has: dimT s r (V) = nr+3 . 3

REMARK 1.2 - Of course one has T?(V) = V*,T$(V) = V . DEFINITION 1.6 - A Euclidean space is an n-dimensional vector space V endowed with a map (scalar product) g : V X V -> R , such that: a) (symmetry) g(vuv2) = flf(v2,vi); b) (nondegeneracy) g(v,v) ^ 0,Vv ^ 0,$r(v,i;) = 0 v = 0; c) (positivityj g(v, v) > 0 , Vv ^ 0 . PROPOSITION 1.3 - 1) If g is a Euclidean structure on the vector space V we get: 9eT°(V).

2) For any basis { e j of V we have: g = gijQ1 u . We often use the convention of not putting the sum-symbol for summations between equal con­ trovariant and covariant indexes.

4

2) We denote the space of symmetric tensors of type (2,0) by SQ(V) . 3) A tensor u^v is said to be skew-symmetric if it satisfies the following condition: u (&) v — —v Q9 u .

4) We denote the space of skew-symmetric tensors of type (2,0) by AQ(V) . PROPOSITION 1.4 - 1) One has the following canonical projections:

(symmetrization)

:

(3 : T 2 (V) - S20(V) \ l \ui®U2*-+

(skew-symmetrization)

} /

- ( U l 0 V>2 + V>2 ® Ml) = Wl 0 «2 J

r a : 2?(V) - Al(V) : < i I wi 0 w2 — i > - ( u i 0 1*2 — u2 0 ^l) = wi A u2

2) One has the following isomorphism :

T02(v)-s02ao®Agoo t h-> s ( * ) @ a(t)

3) If (V,#) is a Euclidean structure, then g belongs to SQ(V) . DEFINITION 1.8 - The symmetry properties can be extended to tensors of type (r, 0) or (0,r). ui 0 - • -0tf r is symmetric if u i 0 - • - 0 u r = u ^ i ) 0 - • -®w ff ( r ), Vcr G 5 r , where 5 r = group of permutations of (1, • • • , r) . The space of symmetric tensors of type (r,0) is denoted by SQ(V) . Similarly, we say that u\ 0 • • • 0 ur is skewsymmetric if: ui 0 • • • 0 ur = e(cr)w(T(1) 0 • • • 0 Wo-(r) , Vcr £ S r , where e( M . 2) The tangent space of M in p G M is the vector space: TpM =

(M,p)^M.

The vectors ofTpM are called applied vectors (in p). 3) If M is a Euclidean space, the corresponding affine space is also Euclidean. 4) We caii dimension of the affine space (M, M, a) the dimension of M . PROPOSITION 1.6 - For any point 0 G M we have the canonical identification: M = M . Furthermore, for any couple (O, {ei})!< n , where {e^} is a basis of M , we have the isomorphism: M = R n . We caii (0,{e{}) an affine frame. Then, to any affine frame it is associated with a coordinate system xa : M —> R , where xa(p) is the a-th component, in the basis {e,} , of the vector p — 0 . THEOREM 1.4 - Let (xa) and (xa) be two coordinate systems corresponding to the following affine frames ( 0 , {e;}) and ( 0 , {ei}) respectively. These are related by an affine transformation: x* = Aapxp + ya (1.3) where (ya)GRn,(A^)e(?%R) DEFINITION 1.10 - We caii A(n) = Gl(n\ R) x R n affine group of dimension n. THEOREM 1.5 - The symmetry group of an n-dimensional affine space is called the affine group A(M) of M . One has the following non-canonical isomorphism: (1.4)

A{M) ~ A(n),

/ ~ ( / ; , y«) ; / « = *« o / = ffi*p + y 7

PROOF. Let us consider the following: LEMMA 1.1 - Let f : M —> N be a differentiable mapping between affine spaces, dim M = ra, dim N — n . The derivative of f is a mapping:

Df :M -► JV(M,N)=

\J

T;M0TqN

^ M x iV x M* (g) N .

p,gGMxN

If {£ a } are coordinates on M , and {t/ J } are coordinates on N , one has the following representation:

Df=

(dxaf)dxa

Y,

®dypof

l E* is also n — k . Then equation A(u) = 0 has k linearly independent solutions { u ( 1 ) , - - - , u W } . The same holds for A*(a) = 0 : { a ^ , • • • , a ^ } . Then equation (1.5) has solutions iff f satisfies the following condition: (1.8)(consistency condition)

< / , a ^ > = 0, 1 < i < k.

If the above equations hold, equation (1.5) has an infinite number of solutions: (1-9)

u=

Y,

CiU^ + Up,

lUi Ui = M, covering of M by means of subsets Ui of M, called open sets; (c) (i : Ui -+ ft,- C R n (Ui)i) = (chart) : < Q,{ = open subset of \ <j>i = bijective

Rn

mapping

_1

(d) the functions fij = { o ^ j : ftj —*• 12, , are of class C fc ,0 < k < oo,o;, where w C means real analytic class. 2) The functions xa = p r a o j : C/"» —* R , where p r a : R n —> R is the projection on the a-th factor, are called coordinates. 3) If n = 1, the manifold is called a curve. 4) If n = 2, the manifold is called a surface. EXAMPLE 1.4 - 1) SPHERE S2 C R 3 . We can obtain a covering by means of six open hemispheres. 2) TANGENT SPACE. TM = [jpeM TPM, where TpM = tangent space at M in p, (TpM is the vector space of the velocities of all curves in M passing through p). One has the covering {Oi}ie/ , where 0 ; = 7r -1 (?7i), with ir : TM —> M the canonical projection, and {Ui}i£i is the covering of M that defines the structure of differentiate manifold on M . We will denote the corresponding coordinates on TM with {xa,ia} , such that xa = xa o TT, xa(v(p)) = va(p), if v(p) = va(p)dxa(p), where dxa(p) is the velocity of the a-th coordinate-line passing through p. In fact, {cfott(p)}i M be a line in a Riemann manifold (M, g). Then, we can induce a metric, i.e., a Riemann structure, on R by means of 7 . More precisely: j*g = 5°(7) o g o 7 : R —► S^R. As a consequence on R we recognize a canonical volume form V(l*g) =

yJgij^xJdt.

2) (AREA OF SURFACE). Let 7 : D C R 2 -► M be a surface on a Riemann manifold (M,g) . Similarly to previous example we can recognize o n D a canonical volume form induced by 7 . More precisely, we have 77(7*2) = VEG - F2du A dv , with

rE = (Gauss-symbols) :

gijidu.j'Xdu.y')

F = 0y(0U.7*')(0t>.V') I G = gii(dv.?)(dv. a&, for any a, 6 G G , and a unity e, ae = ea = a, Va G G , and any element a G G admits an inverse a _ 1 , a a - 1 = a - 1 a = e . Furthermore, the product mapping and taking the inverse, are differentiable of class G*,0 < k < oo,u;. An important example of Lie group is the general linear group GL(n, R ) . In the following we give some other important examples. 2) (GLOBAL STRUCTURE FOR LIE GROUP), (a) Any Lie group G contains a discrete number of unconnected components; the component G 0 connected with the unity is an invariant subgroup, i.e., aGoa~x = Go , Va G G , and the other components are its cosets. In other words, there are as many unconnected compo­ nents as are the elements of the quotient group D = G / G Q that is discrete. 14

(b) G is not necessary a (semi-)direct product [14] of the form G = DxrGo , but this often happens. For example: 0(2n + 1) = Z 2 x S0(2n + 1), 0(2n) = Z 2 xrS0(2n) . (c) The connected component of the unity is the homeomorphic image (i.e., has the same structure of the differential manifold up the class) of a simply connected (i.e., without holes) covering group Go, 7r : G 0 —► Go , such that the kernel of 7r is a discrete group Z contained in the center Z(Go) of Go , namely in the part of Go where the product is commutative. Then, we have: Go = G0/Z . 3) (LIE ALGEBRA OF LIE GROUP). On the tangent space in the unity, TeG , of a Lie group G , we can define a structure of Lie algebra, i.e., a bracket [,}:T e Gx TeG -> TeG,

(ZUZ2)

~

[ZUZ2],

such that the following relations are satisfied: (a)(anticommutativity) [Z1,Z2] + [Z2,Z1)=0,

VZuZ2€TtG;

(b)(Jacobi identity) [Zi,[Z2,Z3]]

+ [Z2, [Z3,Zx]] + [Z s , [ZUZ2]) = 0,

WZUZ2,Z3

G TeG .

In fact, to each vector Z £ TeG we can associate a vector field TG induced by the product of G : " = cr(Z 2 ) a ,l < a < n = dim G . Denote by A(G), (or g , or g) the Lie algebra of G . Then, if { Z a } i < a < n is a basis for A(G), we can write [Za,Zb] = GcahZc, where Ccah G R are called structure constants of G . They satisfy the following relations: 15

(antisymmetry) Cab — ~Cba >

(Jacobi identity) i r~ik

/^ [X, Y] G K . (We write [X, Y] C K.) (b) A subalgebra K of a Lie algebra L is called invariant if [L, K] C K . (c) A Lie algebra L is a direct s u m of two subalgebras: L = K ® M , if this happens for the underlying vector spaces and if [M, K] = 0 . (d) A Lie algebra L is a semidirect sum of two subalgebras: L = K x r M , if [K,L]CK.

I | Alternatively, an invariant subalgebra K C L and its complement M = L — K , form a semidirect sum iff M is a subalgebra. I I The semi-direct sum becomes a direct sum iff both subalgebras are invariant. (e) A Lie algebra L is abelian if [X, Y] = 0 , VX, Y G L . (f) The center Z(L) of a Lie algebra L is the abelian subalgebra such that [Z(L), L] = 0 , Then, we can write: L = Z(L) 0 L0 , where the subalgebra L0 has no center. (g) A Lie algebra is semisimple if it has not invariant abelian subalgebras. (h) A Lie algebra is simple if it is semisimple and does not contain invariant subal­ gebras. | | One can prove that semisimple algebras are direct sum of simple ones. (i)(CARTAN METRIC OF LIE ALGEBRA). On a Lie algebra L we can define a symmetric linear mapping: k e L2S(L) =* L* O L* , such that in any basis we have the following matrix-representation: kab = CacC{;d ,

where Cdac are the constant structures of L . If L is abelian, k = 0 . Furthermore, k can be, in general, degenerate, k is not-degenerate, det k ^ 0 , iff L is semisimple. (1) A Lie algebra is called compact if the corresponding Lie group is such that it can be covered by a finite number of open sets. | | We can prove that if L is compact one can write L = 0 1 < i < s Li , where Li are irreducible compact Lie algebras, where irreducible means simple or 1-dimensional. | | As a consequence, each compact Lie group G can be written as follows: G = Ili rv 7 „ CI/ CJ ,he point £= w ^v\.\ = In particular, at the = 0 one has ^9(*\( ~ S^Sj ^vfii '. 2) The metric on CGij+p is introduced in exactly the same way: the angles are taken in the sense of the Hermitian scalar product in Cl+P [102,161]; (the transformation is replaced by conjugation). Then, Cl+P; with this metric is a Kahler manifold. [101,159] 3) Similar considerations can be made for quaternionic Grassmann manifolds. u. 4) On the manifold Gft,(E) Riemann structure can be induced by using the ,{E) (E) a . +P covering mapping Gt, (E) G +l+p ( £ ) --» -> G,,,+,(S). G G,,,+,(E). £). G+ +p + p(E) M ++p P ((£). *ti G,,, +P

5) In the local coordinates (#/)i