Table Of ContentQuantized
Partial Differential
Equations
A PrAstaro
’:
University of Roma (‘La Sapienza Italy
p:
World Scientific
- -
NEW JERSEY * LONDON SINGAPORE * SHANGHAI * HONG KONG TAIPEI * CHENNAI
CONTENTS
Preface ...................................... vii
Quantized PDE’s I: Noncommutative Manifold ................ 1
1.1 .A lgebraic topology ............................................ 1
1.2 .Q uantum algebras ................................... .... 44
1.3 .Q uantum manifolds .......................................... 63
1.4 .Q uantum supermanifolds ..................................... 88
Quantized PDE’s. 11: Noncommutative PDE’s 103
2.1 .Q uantum PDE’s ..................... 103
2.2 .T he quantum Navier-Stokes equation .... ............... 121
2.3 .Q uantum super PDE’s ...................................... 132
2.4 .T he quantum super Yang-Mills equations ... ... 143
Quantized PDE’s 111: Quantizations of Commutative PDE’s ......... 193
3.1 .I ntegral (co)bordism groups in PDE’s ......... 193
3.2 .A lgebraic geometry of PDE’s ................................ 249
3.3 .S pectral measures of PDE’s ................................. 282
3.4 .Q uantizations of PDE’s ... ... 311
3.5 .C ovariant and canonical quantizations of PDE’s ............. 320
Addendum I: Bordism groups and the (NS)-problem .... 377
4 . 0 .I ntroduction ............................................... 377
4.1 .G eometry and existence theorems for (NS). ................. 381
4.2 .N avier-Stokes boundary value problems ...................... 390
4.3 .T he limit case: The Euler equation .......................... 431
Addendum 11: Bordism groups and variational PDE’s 435
5 . 0 .I ntroduction ............................................... 435
5.1 .H orizontal and vertical forms on PDE’s ...... 436
5.2 .T he category of variational PDE’s ........... 437
5.3 .C onstrained variational sequences ........................... 447
References .... .. .... ... .... 461
Index ............................................................... 473
xiii
PREFACE
A geometric theory of (super) PDE’s for the quantum physics is necessarily
a theory of noncommutative (super)manifolds. In some previous works
we have introduced quantum (super)manifoldst hat appear very useful to
globalize the concept of algebras of quantum operators. For such manifolds
we have developed also a geometric theory of partial differential equations
obtaining theorems of existence of local and global solutions [61-67’69-711.
In some other previous papers we have given also a general proceeding to
quantize PDE’s [58-63,701. In particular we have given geometric methods
to obtain quantizations as covariant and canonical quantization of PDE’s,
that are the most useful from the physical point of view. These methods
are founded on the geometric properties of (super) PDE’s. In this work
we further develop our theory of quantized (super) PDE’s and quantum
PDE’s and explicitly prove the relation between them by means of unifying
theorems that show as covariant quantizations of (super) PDE’s identify
quantum (super) PDE’s in algebraic sense.
A new direction where Mathematics has accumulated important energies
and where we can foresee great results in the next future, is the noncommu-
tative geometry. Really, this is a relatively new area of research in Mathe-
matics, where a central role has been played by A. Connes with his works
published starting from fifteen years ago. But we must include, also, in this
direction the fundamental works by the Bourbaki’s school, the pioneering
algebraic-geometric results by the Rome’s school (C. Procesi), Moscow’s
school (Yu. Manin), Polish’s school and etc. In some sense A. Connes has
vii
viii Quantized Partial Differential Equations
built a bridge between geometry and functional analysis, obtaining impor-
tant results solving interesting geometric problems. These focused, also,
attention by theoretical physicists interested to have mathematical tools to
utilize in their models for quantum physics.
Furthermore, a new direction of research rised in the last seven years in the
framework of noncommutative geometry by us. This new approach develop
a noncommutative differential geometry that uses a point of view nearer to
the classical differential geometry. In fact, it is formulated on pseudogroups
and produces a geometric theory of noncommutative PDE’s, extending in
a natural way, the great results of the commutative geometric theory of
PDE’s. This language is more natural to solve problems in the modern
quantum field theory, where the central role is played by the geometric
structures defining the dynamic field equations. So this new noncommu-
tative geometry offers a new language to solve fundamental problems in
physics. As such problems are of central interest for the scientific commu-
nity, it is expected that, as a push-pull system, all the noncommutative
geometry should be more considered by the mathematical community. Let
us emphasize that our geometric theory of quantum PDE’s is the natural
development , in noncommutative sense, of the geometric theory of PDE’s.
Such a theory, just starting from the pioneering works by the European-
American school (E. Cartan, H. Goldschmidt, V. Guillemin, M. Kuraniski,
B. Malgrange, D. Spencer, S. Stenberg, W. J. Sweeny,. ..) is the unique ap-
proach that has obtained very general theorems and great results in the the-
ory of PDE’s. The actual soldering of the Cartan’s theory with the formal
theory, (obtained principally by the Moscow’s school), from one side and the
more recent results (by A.PrBstaro) extending algebraic-topological points
of views, ((co)bordism groups), to the previous geometric theory of PDE’s,
from another side, have given to us the opportunity to obtain very general
local and global informations on the geometric structure of PDE’s and their
solutions. These results should also convince analysts that geometric meth-
ods are fundamental also in the analysis of PDE’s, both commutative or
noncommutative. In fact, are just these geometric methods that have given
to us the opportunity to solve very crucial open problems in the theory
of PDE’s. Such are, for example, the existence of global (smooth) solu-
tions for the (quantum) Navier-Stokes equation and the (quantum) (super)
Yang-Mills equation. For the last equation a criterion to recognize global
Preface ix
solutions with mass-gap were also obtained.
The aim of this work is first to report on such a new theory and them to
relate it with a previous formulation of quantized PDE’s developed starting
from fifteen years ago. This formulation of quantization of PDE’s is just
founded on the geometric structure of PDE’s and gives us a fully covariant
description of the covariant and canonical quantizations. (These are the
most interesting, from the physical point of view, between all the possible
quantizations of PDE’s and are shortly referred as Dirac quantizations also.)
Note that a purely algebraic description of quantum phenomena, i.e., by
simply using noncommutative algebras, is not satisfactory as, in this way,
we are not more able to recover the meaning of full covariance that, instead,
is fundamental for a good description of any natural geometric theory and
hence of any physical theory. Our approach to quantum (super)manifolds
overcomes this difficulty as uses the noncommutative algebras as a ”fun-
damental algebra of noncommutative numbers”, with which to build the
models for quantum manifolds. In this way we can naturally extend the
classic commutative geometry to a noncommutative context and give also
a natural noncommutative meaning of full covariance. We explicitly prove
that Dirac covariantly quantized PDE’s identify quantum PDE’s. So the
ring is closed! We can start from a classical theory, i.e., a commutative
PDE, canonically quantize it and obtain a quantum PDE. To this last we
can apply all our results of the geometric theory of quantum PDE’s and
obtain a full characterization of our quantum problems.
This work is a collection of three long papers (part. I, part. 11, part. 111)
devoted to the geometric theory of noncommutative manifolds, noncom-
mutative PDE’s and quantized PDE’s. The category of noncommutative
manifolds is that of quantum manifolds as introduced in some our previous
works. The aim is to give a formal relation between classical, quantum
and quantized PDE’s in a unified geometric building. Therefore, this work
is divided in three parts. Part I. The first section of this part is devoted
to look through some definitions and results in algebraic-topology that are
used in all the book. The other three sections are reserved to the de-
scription of some suitable categories of noncommutative manifolds called
quantum manifolds and quantum supermanifolds respectively. The starting
points for such structures are quantum algebras and quantum superalgebras
respectively. These are particular classes of topological noncommutative
X Quantized Partial Differential Equations
algebras, that are useful to locally interpret the concept of quantum fields.
The globalization of such structures is made by means of quantum (su-
per)manifolds that are built by means of suitable pseudogroups on suitable
model spaces. In this context fundamental is the role of the class of dif-
ferentiability that we call quantum differentiability. Then we characterize
quantum (super)manifolds by their differential and algebraic-topological
properties. The emphasis is put on their (co)homologic and (co)bordism
groups. This is made in order to obtain suitable tools to use in the geometric
theory of (quantum) PDE’s. Part 11. Here we develop the geometric theory
of quantum (super) PDE’s obtaining general theorems that allow us to state
existence of local and global solutions. Global properties of solutions are
characterized by means of integral (co)bordism groups of quantum (super)
PDE’s. Furthermore, we apply such results to some important equations
of the mathematical physics. Particular attention is devoted to the quan-
tum Navier-Stokes equation and quantum Yang-Mills equation. For such
equations we obtain also theorems of existence of global solutions, solving
well-known open problems under focus of the mathematical community.
Part 111. In the first section we consider the geometric theory of PDE’s and
their characterizations by means of integral (co)bordism groups, following
our previous results on this subject. In section two an algebraic theory of
PDE’s is given also. In section three a theory of spectral measures of PDE’s
gives us the way to interpret the meaning of quantizations of PDE’s on the
ground of the mathematical logic, that is just the subject of section four.
Furthermore, in section five the specialization on the covariant and canon-
ical quantization of PDE’s is made by using purely differential geometric
techniques. Here we resume some our previous results on the quantizations
of PDE’s and give also some further results on the covariant and canonical
quantizations of PDE’s. In particular, we prove that Dirac covariant quan-
tizations of (super) PDE’s identify quantum (super) PDE’s. Therefore,
for such quantized equations our theory of quantum (super) PDE’s can be
applied. Many applications are given to equations describing important
phenomena of quantum physics also.
Let us emphasize also that the geometric setting to obtain covariant and
canonical quantizations of classical PDE’s interesting field theory, realizes
a bridge between noncommutative PDE’s and classic PDE’s. This just
justifies our use of the word ”quantum”, according to its original physical
Preface xi
meaning for which has been first forget.
Finally two addenda, one devoted to the relations between integral bordism
groups for the Navier-Stokes equation and the solution of the fundamen-
tal problem on the existence of global smooth solutions of this important
PDE (Addendum I), and another that studies the relations between inte-
gral bordism groups and constrained variational problems (Addendum 11),
conclude this book.
Roma, Easter 2003.
Agostino Prastaro
QUANTIZED PDE’s I:
NONCOMMUTATIVE MANIFOLDS
Abstract - In this first part we consider quantum (super)manifolds as topologi-
cal spaces locally identified with open sets of some locally convex topological vector
spaces built starting from suitable topological algebras, quantum (super)algebras.
The noncommutative character of such quantum (super)manifolds is given by the
underlying noncommutative algebras A. In fact, here A plays the role of ”funda-
mental algebra of numbers”, like K=R,C does for usual commutative manifolds.
Therefore, quantum (super)manifolds are the natural generalization of manifolds,
when one substitutes commutative numbers with noncommutative ones. Commu-
tative manifolds are contained into quantum (super)manifolds, as quantum (su-
per)algebras A are required to contain K. This aspect is also reflected by the fact
Qk
that the class of differentiability for pseudogroup structures defining quantum
(super)manifolds contains the usual Ckd ifferentiability for manifolds. In fact, the
class of differentiability of such topological manifolds is defined by requireing weak
differentiability and 2-linearity of the derivatives, where 2 is the centre of the
underlying quantum (super)algebras. We give (co)homological characterizations
of quantum (super)algebras and quantum (super)manifolds by applying to these
noncommutative topological manifolds standard methods of algebraic topology. In
particular, we calculate also (co)bordism groups in quantum (super)submanifolds.
1.1 - ALGEBRAIC TOPOLOGY
As in this work we systematically use the language and tools of the alge-
braic topology, and even if we suppose the reader just introduced in this
subject, we consider suitable to start with some fundamental definition and
results on the algebraic topology, emphasizing those that will be used in
the following. (See also, e.g., refs.[17,77,81-84].)
In all this work we will denote by K be the field of real numbers R or
complex numbers C. Moreover, we denote by N the set of natural numbers
and by Z and Q the fields of rational integers and rational numbers respec-
tively. We will denote by H and 0 respectively the algebras of quaternions
and octonions.
1
2 Quantized Partial Differential Equations
Here U denotes disjoint union and I the segment I = [0,1] c R. We
will work in the category loopf t opological spaces and continuous maps.
For any X 6 Ob(zp),w e set X+ = X/@ = XU{*}, where * denotes
a point. Therefore, X f is the compactijied to a point of the topological
space X . Two topological spaces X and Y that are homeomorphic will be
=
denoted by X Z Y. We shall, instead, write X Y if they are homotopy
equivalent, i.e., there exist two maps f : X --f Y,g : Y -+ X , such
= =
that g o f lx and f o g ly, i.e., there exist continuous mappings
(homotopies) F : X x I 4 Y and G : Y x I -+ X ,s uch that Flxx(o)= go f,
Flxx(l) = lx, Glyx{O) = f 09, G I Y ~ {=~ l)y . The set of all homotopy
classes of maps X + Y is denoted by [X,Y].I f A c X is a subspace,
a homotopy F : X x I -+ Y relative to A is such that F(a,t ) = F(a,O ),
'v'u E A, t E I. If Fo = f and F1 = g, we write f =A g. We say also that
F is a homotopy re1 A. In the following table we resume some fundamental
operations in algebraic topology.
-
TAB.l.l Fundamental opertions in algebraic topology
I I
Name Definition
I I
cone of X CX=Xxl/Xx (1)
suspension of X SX~XXl/{XX{O} } U{XX{1 ))
wedge product of (X,a) and (Y,b) XVY=Xx{b} U/Yx{a}
smash product of X and Y X AY X x Y/( XV Y)
One has the natural equivalence: [SX,* ; Y,y o] ++ [XZ,O; RY, WO],w here
RY is the loop space of Y. This is an isomorphism of groups. One has
the following one-to-one correspondence: [CX,* ; Y,y o] [XIz o; PY, wo],
++
where PY ZE (Y,y ~)('~~Th)e. map p : PY + Y,p (w) = w(1) is continuous.
One has p-l(y0) = RY.
Definition 1.1. For any pointed topological space (X,zo) we define nth
homotopy group (or nth Hurewicz group) 7rn(X,z o) = 7ro(R"X,wo).
Proposition 1.1. One has the following isomorphisms:
=
7rn(X,zo)2 [S1,So;Rn-l,wo]2 [52,So;R~-2,wo] [Sn,SrJ;X,ZO].
''_
A pointed space (XI5 0)i s called contractible if it has the same homotopy
type of ({xo},~~)T.h en, if (X,ZO)is contractible, then 7rITn(X,zo=) 0,
n 2 0.
