Table Of ContentQuantum field theory: the bridse between
mathematics and the physical world
by
Peter Braam
0 § Introduction.
These notes, based on a talk on the subject, should serve one purpose only:
make mathematicians enthusiastic for quantum field theory. Thus they have
been written in a heuristic rather than a completely rigorous style, in
fact many topics mentioned in 2 § have yet to be rigorously formulated.
We have stressed the central role that symplectic geometry plays in the
theory, and the analogy between ordinary quantum mechanics and the quantiz-
ation of field theories. Therefore, in section 1 classical mechanics is
rather quickly repeated, and in section 2 quantization is treated. We would
like to thank Dr. P. v. Baal for long discussions on the subject.
1 § Mechanics and momentum maps.
The importance of mechanics lies not only in its physical origin, but also
in the fact that mechanical ideas are widespread in physical quantum theo-
ries and in mathematics. To describe a mechanical system one can choose a
certain setting, each with its own advantages. Here the following two are
used:
)I Lagrangianf0rmulation
Motion is a path, c(t) , in a confisuration manifold M , and the path
which is actually followed by our mechanical system is a critical point
of the action functional,which is a map from the space of paths to • ,
defined by:
S(c) = J L(c(t) , &(t))dt I-I
where L : TM~ is the Lagrangean. Now L is not an arbitrary element
of C~(TM) and we require it to be of the form
L(m, m) = X gm(Xm, m) X -V(m) 1-2
where g(.,.) is a Riemannian metric on M , called kinetic energy, and
V: M~ is the potential. The critical paths c(t) are solutions of the
Euler Lagrange equations:
-- ~ (c(t), ~(t)) = -- (c(t), ~(t)) j = i ..... n 1-3.
~t ~qj ~qj
Here (ql' "'" ' qn ) are local coordinates around c(t) on M .
The physical meaning of the action functional may seem mysterious at this
stage, but quantum mechanics sheds some light on this, see §2, and Manin
[9], Ch. .3
Before we proceed to give the Hamiltonian or symplectie formulation, we
discuss an example:
Celestial mechanics. Consider n particles in ~3 with masses m. .
i
The configuration space is M= R 3n and let qki be coordinates on M
with k= 1,2,3 , i= ,i .... n . Then:
L(q, q) = ~ i
~m i lliqll 2+ ~ Gmim. llqi-qjll -I
i i<j J
1-4
2miq~i + I gm. m. ~ (qki-qkj)2
k,i i<j i j k=l
( G>0 , the gravitational constant). The first term on the rhs of (1-4)
clearly represents the kinetic energy, whereas the second is the gravita-
tional potential. Most facts in mechanics were discovered solely through
consideration of the equations of celestial mechanics:
(q~i - q-') i m G m.
d .. = _ ~ K K~ ~ I = i .. n
d-tmi qki = miqki i<j f ~ ~ ~ (q%i-q~j)2\3/2j . . . . .
the Euler-Lagrange equations of (1-4).
2) Hamiltonian formulation
The advantage of Hamiltons version is that the place (qi) and speed (~j)
variables no longer play a different role. Put differently the theory is
mathematically a bit more intrinsic.
Let X be a symplectic manifold , i.e. a manifold X with a so-called
symplectic form, £ ~ ~2(X) , where ~ obeys
a) ~x : T X × T X ~ R~ is nondegenerate
X X
b) dm = 0 .
A mechanical system is now described by a Hamiltonian or total energy func-
tion C c H (X) . This Hamiltonian H determines a vectorfield H V on X ,
called the Hamilton vectorfield of H , defined as follows:
i ~ = dH (i is contraction) 1-5
H V
Hami!tons equations now read
~(t) = VH(X(t)) . 1-6
If we put = X T*M then we can recover the Euler Lagrange equations. For
convenience we work in coordinates qi on M . The conjugate momenta:
~L = ~g(~,~)
Pj = __
~qj ~qj
supply coordinates on T*M X = and it is not hard to see that
w = ~ dpj ^ dqj
J
defines (globally~) a symplectic form on X . Associated to a Lagrangian
L(q,~) is the Hamiltonian
H(q,p) = ~pj ~j (p) -L(q, ~(p))
J
Now the Hamilton vector field is given by:
~H ~H .\
VH(q,p) = (+ ,
so Hamiltons equations read:
~H
~p
1-7
~H
~q
The equivalence of 1-7 and 1-3 follows easily from the definitions of p
and H . A more extended exposition of Hamilton's and Lagrange's descrip-
tion of mechanics can be found in Abraham-Marsden [i] and Arnold [3].
Finally we discuss conserved quantities in mechanical systems in the set-
ting of momentum maps, see also Bourguignon's seminar notes. Let G be a
connected Lie group acting symplectically on (X,~) ; this means g*~ =
c (g )G . Consider the vectorfields induced by the action, i.e. consider
the map ( g is the Lie algebra og G )
~: g ~ F~X): Y ~ Y = {m ~ ~t (exptYm),t=0 }
Then for Y~ g , L~ = d(i~)= 0 , so assuming for simplicity that
HI(x, ~) 0 = , we get:
i~ = d ~y for some ~y£ C~(X)
in other words ~ is the Hamilton vectorfield of #y . Now ~y is only
determined up to a constant, but with a bit more work we get a map:
¢ : X ~ g* s.t. <~,Y> = ~y , Ye g
and: 1-8
Cy(g.x) = ~Adg_l y(X) gc G
( ~ is almost unique, see Abraham-Marsden ]I[ Ch. 4). ~ , first defined
by Lie, is called the momentum map of the G action, and ¢ determines
the complete G action by integration of the Hamilton vectorfields
= V~y
From 1-8 we see that the co-adjoint orbit of ~(x) c g~ is constant on the
orbit of x~X (because ~(gx) = tAdg- I ~(x) ). If G is Abelian we get
that ~(x) is constant on Gx . In general ~-I{0} is a G invariant set
and
~-i{0} G / 1-9
is called the symplectic quotient or Marsden-Weinstein reduction of
X: ~-i{0} G / is symplectic itself.
Example: Given a Hamiltonian H on (X,~) whose flow is complete. Here
acts symplectically by:
(t,x) ~ ~(t,x) X ~
where ~(t, ) is the flow of H . V Clearly for g e = 1 ~ we have ~=V H
so
: X ~ ~* : x ~ H(x)
is constant on the ~ orbits. Here we recovered conservation of energ[.
2 § Classical field theories and their quantization.
In §I classical mechanics was presented. Here our interest lies in the quan-
tization of this, leading to quantum mechanics. Next we introduce classical
field theories, more specifically gauge theories. These can be quantized
just as classical mechanics, but the original, classical, Hamiltonian sys-
tem is now infinite dimensional. The theory has not yet found a mathemat-
ically mature framework; our considerations will be heuristic.
To quantize classical mechanics we give a scheme of geometric quantization
(see Abraham-Marsden ]i[ 5.4). The arrows ~ mean something like "corre-
sponds to".
Classical mechanics Quantum mechanics
)i( Points (q,p) eX k = T*IR Functions ~ in Hilbert space
e2(~ k) , called wave functions.
More precisely in symplectic terms:
Points in a symplectic ~ Sections of a - ¢ line bundle
manifold ~) (X, L with connection V, which
are coy. constant on the leaves
of a Lagrangean foliation of X .
This process is called geometric quantization.
(ii) Certain functions on X Self adjoint operators on
L2(~ k) called observables.
Example:
place functions qi ~ ~ ~(x) ~ xj ~(x)
momentum " pj " " ~(x) ~-- i~ ~
J
Here is a small positive number called Planck's constant. The
limit ~0 should describe classical mechanics.
)iii( paths q(t), p(t) A function (x,t) ~ on k IR×]R
s.t. ~(-,t) eL2(~ )k
initial values p(0), q(0) c A function k) L2(]R ~(-,t) E
llamiltons equations in a )a Schr~dingers equation:
~otential V(x) i~ ~-~ = ~62mA + ~ V(x)~
b) The proPagator or Greens
function K defined by:
~(x,t) = [ K(x,t; y,0) ~(y,0)dy
)a and )b are equivalent: see below~
So far our scheme. The idea of quantum mechanics is that physics is de-
scribed better using the quantum theory than the classical theory. The wave
function ~(x,t) si supposed to describe the system, not really the states
of the system; these aren't so directly accessible. Information of a prob-
~ilistic kind about the states can be obtained throu~ the observ~les,
e.g.:
)i( Expectation values of coord, jq at time t :si
I ~(x,t) xj~(x,t) dx = <~,Qj ~>L2 •
(ii) Heisenberg's uncertainty relation: a( consequence of the set up)
(variance in observed) fvariance in observed)
values of jq " \values of pj ~ ~
Note that in particular (ii) shows a drastic difference with classical
physics. Further details of quantum mechanics can be found in Dirac's book
[5].
Before we turn to classical field theories we discuss the propagator, fol-
lowing Lee ]8[ ch. .91 The propagator is of course the heart of quantum
mechanics because ti describes the dynamics.
First write Schrodinger's equation in the form:
I ~ )@(H )~(H
= = ~2m A ~ + V(x)#
~t ~
It follows that
¢(-,t) = exp(tH/~i) ¢(-,0) ,
so K(-, ;t ,- )0 si just the integral kernel of exp(tH/~i) . Feynman,
following a suggestion of Dirac, has given an explicit formula for
K(x, ;t y, )0 in terms of a path integral:
K(x,t;y,0) = I exp(iS(q(s))/~l dq(s) 2-1
paths q(s)
q(0)=y, q(t)=x
In this formula S(q(s)) is the classical action of the path q(s) and
dq(s) is a so-called Gauss-Wiener measure on the space of paths. (Wiener
had derived a similar formula for the heat equation earlier).
Integrals if this kind have been defined rigorously, see Simon [Ii], but we
take a heuristic view at them, using a finite dimensional approximation of
the space of paths. Pick an Ne ~ , put k= 1 for simplicity, and approx-
imate a path q(s) by the "step-path" (fig. 2,1):
/ :
fig. 2.1
N x
t
o , o , o ~ o e
qO=Y qN-1
• & ,.
o 2tl t
Now (2-1) can be computed :sa
lim N C I" exp [~ t ~ {im( jq . -tq/N j-i 2 ) - v(qJ + 2 qj-I dql )}] "'" dqN_ I
N ~- ~ ]RN-I j =I '
2-2
with N = C (2wi~t/(N-l)m) -(N-I)/2 A . fancy proof that K(x, t;y,0) is
indeed the kernel of the Schr~dinger propagator can also be given along the
line of a step path argument. Remark that in (2-2) the integrals are never
absolutely convergent~ According to some physicists, the advantage of using
the Feynman integrals instead of the Schrodinger equation ,si that they con-
tain only the action, which is usually (but not here) a Lorentz invariant
object.
Next we introduce gauge theories in vacuum. Let ~3,1 be Minkowski space,
G a compact Lie group and P a principal G-fibre bundle over R] 3'I . The
configuration space here is not a space of paths but the space C(P) of
connections or gauge potentials on P . The action YM: C(P) ~ is called
the Yang-Mills functional and is defined by
MVdMIIAFII
YM(A) = ~IR3'1 2-3
where the subscript M denotes "Minkowskian", and
A = F dA + I [A,A] 2-4
is the curvature or field strength of A . The corresponding Euler Lagrange
equations are the so called Yang-Mills equations:
dA*M FA = 0 . (*M is Minkowski Hodge star). 2-5
The Yang-Mills functional is invariant under the action of an infinite di-
mensional Lie group GA(P) on C(P) . This group is called the gauge group
and defined by:
GA(P) = F(P XAdG)
A gauge transformation gc @A(P) sends Ae C(P) to
g- A = Lg, dg -l+AdgA 2-6
This implies F gA = Ad A F and explains invariance of the action,
g
Example: Maxwell's electromagnetism in vacuum. Here G=U(1) so A and
F are up to a factor i just ordinary forms. As A F =dA we get
A = d d F dA = 0 2-7
d*F A = d*dA = 0 2-8
Writing dA = .E dx i ^dt +B.. dx i ^dx j it is easy to see that up to physic-
l jl
al constants 2-7 and 2-8 are the four Maxwell equations in vacuum. The
action of g = e i~ c GA(P) on A is given by
-i
g(iA) = gd-gI + giAg = i(A-d~)
and this provides the principle that adding "a gradient" doesn't affect
electromagnetism.
To achieve quantization of a gauge theory, i.e. to get a quantum-field-
theory, we first switch to a Hamiltonian picture. For simplicity we restrict
ourselves to electromagnetism, but the discussion can readily be generalized,
see Arms [2] and Itzykson Zuber [7] 9-3.
Let C be the space of gauge potentials on ~3 , so C=~I(~3) . Then
T*C ~ ~i(~3) x ~2(~ )3 via the integral pairing. For a curve A(t) in C
we define an action
S(A(t)) = fL(A(t), A(t))dt 2-9
with
i • I
L(A, A) = ~IIAII~2,~ - 3 ~ L2lIAdil 2,~3 2-I0
If we let the path A(t) in C correspond to the ~3,1 gauge potential
A(x,t) = ~*A(t) , with ~ : R3, I~3 the space projection then 2-9, 2-10
is precisely the Yang-Mills action 2-3, YM(A) . The electric and Magnetic
field can be found as: E=A(t) , H =dA . The conjugate momentum of A(t) =E
is the so-called electric displacement:
D = *E = *A(t) (* the ~3 Hodge star)
so this gives rise to a Hamiltonian H :
H: T*C ~ ~: (A,D) ~ H(A,D) = ~ Dil + lldA
2 2 2
The corresponding Hamilton equations are:
= *D 2-9
= d*dA = d*H = dB = rot B 2-10
Now 2-9 results in dA = d*D = rotE , a Maxwell equation; 2-10 is another
and a third can be found by noting that ddA = dH = divH = 0 , but divD=0
is missing.
To find this missing sheep, we note that GA = C~(]R 3, U(1)) acts in a Ha-
miltonian way on T*C as follows:
10
i~, (e (A,D)) ~- (A-d~, )D eT*C
The momentum map : ~ T*C ~- ga* --~ ~3(IR )3 is given by
(A,D) ~ dD = divDc~3(]R3) ~= ga* .
So we should work on ~-I{0} to get the fourth Maxwell equation divD=0
satisfied. However, to quantize we need a symplectic manifold and this can
be achieved rather naturally now by going over to the Marsden Weinstein re-
duction Cphy = s ~-i{0} /CA ° ( GA ° component of i of GA)*. So what we
see is that the constraint 0 = ~ on our Hamilton equations, naturally brinss
with it that only sause invariant objects should be considered in doing
quantum theory. This is an important observation in building up a quantum
field theory, see Itzykson Zuber [7] 9-3.
We may now take a look at the Feynmann integrals for a gauge theory. From
2-1 we see that they must look like:
f exp{iS([A](t))}d[A](t)g . 2-11
paths in ~-I{o}/GA~
with boundary cond.J
here g is a constant >0 , called the coupling constant. Now an important
result, the so-called Ward identities, that 2-11 can be written as:
I exp { gYM(A)} d[A] 2-12
connections on [0,t]
with boundary cond. x RI 3} / GAdi 4 m
i.e. as an integral over connections on a four manifold modulo the gauge
group in dimension four. The derivation of 2-12 from 2-11 involves some cal-
culations using 6-functions of the set {~=0} , see L. Faddeev [6]. The
Euclidean postulate now states that 2-12 must be computed by replacing t
by - iT . From 2-2 and 2-3 one easily deduces that this amounts to re-
placing the Minkowski "norm" of A F by the Euclidean one, and here is the
birth of the Euclidean gauge theorie_s. For the Euclidean postulate see It-
zykson, Zuber ]7[ 6-2, there it is referred to as Wick rotation.
* We have omitted the o from GA in the sequel for simplicity.
o
