Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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r and r
0
and different but cohomologous formal
series of 2-forms and
0
, one has to compare the
corresponding Fedosov derivations D and D
0
. First
recall some well-known facts about torsion-free
symplectic connections on (M, !). Given two such
connections r and r
0
, it is obvious that
S
rr
0
(X, Y):= r
X
Y r
0
X
Y, where X, Y 2
1
(TM)
defines a symmetric tensor field S
rr
0
2
1
(
W
2
T
M TM)onM. Defining
rr
0
(X, Y, Z):=
!(S
rr
0
(X, Y), Z) it is easy to see that
rr
0
2
1
(
W
3
T
M) is a totally symmetric tensor field.
Conversely, given an arbitrary element 2
1
(
W
3
T
M) and a symplectic torsion-free connec-
tion r and defining S
2
1
(
W
2
T
M TM)by
(X, Y, Z) = !(S
(X, Y), Z), then r
defined by
r
X
Y := r
X
Y S
(X, Y) again is a torsion-free
symplectic connection and all such connections can
be obtained this way by varying . Using these
relations, one can compare the corresponding
mappings r and r
0
on W. With the notations
from above we have
rr
0
¼ðdx
j
dx
i
Þi
s
ðS
rr
0
ð@
i
;@
j
ÞÞ
¼
1
adðT
rr
0
Þ½29
where T
rr
0
2
1
(
W
2
T
M T
M) W
1
is
defined by T
rr
0
(Z, Y; X):=
rr
0
(X, Y, Z) =
!(S
rr
0
(X, Y), Z). Moreover, T
rr
0
satisfies the
equations
T
rr
0
¼ 0 ½30a
and
rT
rr
0
¼ R
0
R þ
1
T
rr
0
T
rr
0
r
0
T
rr
0
¼ R
0
R
1
T
rr
0
T
rr
0
½30b
where R = (1=4)!
it
R
t
jkl
dx
i
_ dx
j
dx
k
^ dx
l
and
R
0
= (1=4)!
it
R
0t
jkl
dx
i
_ dx
j
dx
k
^ dx
l
denote the
corresponding elements of W
2
that are built
from the curvature tensors of r and r
0
.
Now we are in the position to compare two
Fedosov derivations D and D
0
resp. the induced star
products and
0
obtained from (r, ) and (r
0
, ).
The idea for the construction of an equivalence
transformation from to
0
is to look for an
automorphism A
h
of (W, ) of the form
A
h
¼exp
1
adðhÞ
such that D
0
¼A
h
D A
h
ðÞ
1
½31
where h is an element of W
3
guaranteeing that A
h
is
well defined and without loss of generality is assumed
to satisfy (h)=0. In case one can find such an
element h it is clear that A
h
yields a bijection between
ker(D ) \W and ker(D
0
) \W and hence one would
obtain an equivalence S
h
from to
0
defining
S
h
f :¼ ðA
h
ð f ÞÞ
¼ exp
1
adðhÞ
ð f Þ
½32a
with inverse
ðS
h
Þ
1
f ¼ ðA
h
ðÞ
1
0
ðf ÞÞ
¼ exp
1
adðhÞ
0
ðf Þ
½32b
A direct computation yields
A
h
D A
h
ðÞ
1
¼ D
1
ad
expðð1=ÞadðhÞÞ id
ð1=ÞadðhÞ
ðD hÞ
which is equal to D
0
iff h has been chosen such that
T
rr
0
þr
0
r
exp
1
adðhÞ
id
1
adðhÞ
ðD hÞ2W
1
½33
is a central element. Considering the total degree of
the terms in this expression, this is the case iff there
is a formal series of 1-forms C 2
1
(T
M)[[]]
such that the expression in eqn [33] equals 1 C.
Applying D to this equation and using the
equations that r and r
0
satisfy together with the
relations [30] it is cumbersome but not difficult to
show that necessarily and
0
have to be
cohomologous:
0
¼ dC ½34
with C as above. Now, using [6] one can show that
this condition is in fact sufficient and moreover one
can even determine the element h in question
recursively:
Theorem 5 (Fedosov 1994, theorem 4.3). Two
Fedosov star products and
0
obtained from (r, )
and (r
0
,
0
) are equivalent iff and
0
are
cohomologous. In case C 2
1
(T
M)[[]] satisfies
0
= dC there is a uniquely deter mined element
h
C
2W
3
with (h
C
) = 0 such that
T
rr
0
þr
0
r
expðð1=Þadðh
C
ÞÞid
ð1=Þadðh
C
Þ
ðD h
C
Þ¼1 C
Moreover, h
C
can be determined recursively from
h
C
¼C 1 þ
1
rh
C
1
adðrÞh
C
ð1=Þadðh
C
Þ
exp ð1=Þadðh
C
ÞðÞid
r
0
r þT
rr
0
½35
296 Fedosov Quantization
and with the so-constructed h
C
one has
D
0
=A
h
C
D (A
h
C
)
1
and thus S
h
C
according to eqn [32]
defines an equivalence transformation from to
0
.
Evidently, in the above construction of the
equivalence transformation S
h
C
there is some choice
of the formal series of 1-forms C. Different possible
choices C and
e
C differ by a formal series of closed
1-forms but choosing
e
C instead of C amounts to
another equivalence transformation S
h
~
C
= A
0
S
h
C
A
from to
0
, where A and A
0
are certain self-
equivalences of and
0
, respectively. In case and
0
are real, we have seen that as well as
0
are
Hermitian star products and it is easy to verify that
choosing a formal series C of 1-forms as above that
is moreover real yields an element h
C
satisfying
h
C
= h
C
. But then it is evident that the resulting
equivalence transformation is also compatible with
complex conjugation, that is,
S
h
C
f = S
h
C
f for all
f 2C
1
(M)[[]].
Now we are prepared to give a construction of all
C[[]]-linear automorphisms of a Fedosov star
product . It is easy to show that any C[[]]-linear
automorphism of a star product ? on a symplectic
manifold is the combination of the action of a
symplectomorphism : M !M and an equivalence
between ? and the pullback ?
0
via
1
of ?, which is
defined by f ?
0
g = (
1
)
((
f ) ? (
g)) (cf. Gutt and
Rawnsley (1999 Proposition 9.4)). Since the char-
acteristic class of ?
0
is given by c(?
0
) = (
1
)
c(?), the
necessary and sufficient condition for a
symplectomorphism to define a possible zeroth-
order term of an automorphism is that (
1
)
c(?)
= c(?) since ?
0
and ? have to be equivalent.
Within Fedosov’s framework, it can be shown
that the pullback
0
via a symplectomorphism
1
of
is identical to the Fedosov star product obtained
from (r
0
= (
1
)
r
,
0
= (
1
)
), which just
expresses the functoriality of Fedosov’s construc-
tion. Together with Theorem 4 this particularly
shows that c(
0
) = (
1
)
c(), and therefore
0
is
equivalent to iff and
0
differ by a formal series
dC
of exact 2-forms, where C
2
1
(T
M)[[]]
clearly depends on . But in this situation one can
apply the construction of equivalence transforma-
tions between Fedosov star products given in
Theorem 5 with C replaced by C
and r
0
,
0
as
above yielding an equivalence S
h
:= S
h
C
from to
0
.
Finally, we therefore get that the combination
A
:¼
S
h
½36
is a C[[]]-linear automorphism of and it is
obvious from the above that every such automorph-
ism can be obtained by considering all symplecto-
morphisms of (M, !) satisfying [(
1
)
] = [] and
composing the resulting A
according to [36] with
all self-equivalences A of according to [28].
Adaptions, Modifications,
and Generalizations
The geometrical construction of Fedosov has gone
through many adaptions and modifications that are
well suited to the particular geometry of the under-
lying symplectic manifold. Moreover, there are
generalizations that go beyond the case of symplec-
tic manifolds and others that yield more general
deformations than star products. We just give a few
important examples that stress the power and
beauty of Fedosov’s construction.
On a Ka¨ hler manifold, one can define the notion
of star products with separation of variables (cf.
Karabegov (1996) that are also called star products
of Wick type (cf. Bordemann and Waldmann (1997)
and Neumaier (2003). These are star products such
that in local holomorphic coordinates the bidiffer-
ential operators C
r
are of the form
C
r
ðf ; gÞ¼
X
K;L
C
K;L
r
@
jKj
f
@z
K
@
jLj
g
@
z
L
½37
with certain coefficient functions C
K; L
r
.Thesestar
products can be obtained by a modified Fedosov
construction starting from the product
Wick
on W
given by
a
Wick
b¼ exp
2
i
g
k
l
i
s
ð@
z
k
Þi
s
ð@
z
l
Þ
ða bÞ
½38
where g
k
l
denotes the components of the inverse of
the Ka¨ hler metric in local holomorphic coordinates.
In the case of a Ka¨ hler manifold, there is a
distinguished torsion-free symplectic connection
namely the Ka¨hler connection r that induces a
superderivation of
Wick
in a way completely
analogous to [8]. With these structures the Fedosov
construction works for an arbitrary formal series
of closed 2-forms as before, but one can show that
the resulting star product is of Wick type iff is of
type (1, 1) and one can even show that one obtains
all star products of Wick type by varying
(cf. Neumaier (2003)).
In the case of an almost-Ka¨hler manifold, one can
consider a product
0
on W similar to
Wick
which is adapted to the almost-complex structure
(cf. Karabegov and Schlichenmaier (2001)). How-
ever, in this situation there is no torsion-free
connection that yields a superderivation of this
product but only a connection r
0
with torsion that
defines such a superderivation. Nevertheless, one
Fedosov Quantization 297
can consider a generalized Fedosov construction. To
this end, one shows that [, r
0
] = (1=)ad
0
(T
0
) with
some T
0
2W
2
that satisfies T
0
= 0 and encodes
the torsion of r
0
and R
0
= r
0
T
0
, where again
r
02
= (1=)ad
0
(R
0
) and R
0
, which depends on the
curvature of r
0
, satisfies r
0
R
0
= 0. But then it is easy
to show that there is a unique element r
0
2W
2
1
such that
r
0
¼r
0
r
0
1
r
0
0
r
0
þ T
0
þ R
0
þ 1
and
1
r
0
¼ 0 ½39
with as above, which can also be computed
recursively. Clearly, D
0
= þr
0
(1=)ad
0
(r
0
)then
is a suitable Fedosov derivation with square zero for
0
and one can proceed as described earlier to obtain a star
product
0
adapted to the almost-complex structure.
On a cotangent bundle : T
Q !Q, where T
Q
is equipped with the canonical symplectic form
!
0
= d
0
, one can consider (cf. Bordemann et al.
(1998)) the following so-called standard ordered
product
std
on W given by
a
std
b ¼
exp i
s
ð@
p
i
Þi
s
ð@
q
i þp
l
l
ik
@
p
k
Þ
ða bÞ
½40
in local Darboux coordinates. Here
l
ik
denotes the
Christoffel symbols of a torsion-free connection r
Q
on Q in the chart of Q corresponding to the bundle
chart (q,p) and it is straightforward to see that
std
does not depend on the chosen local coordinates and
is associative. In the present situation, one can
define a torsion-free symplectic connection r
T
Q
on
T
Q solely in terms of r
Q
but then the correspond-
ing mapping r
T
Q
on W again fails to be a
superderivation of
std
, whereas the combination
r
T
Q
þB with B=(=3)p
l
R
l
jik
(1 dq
i
)i
s
(@
p
j
)i
s
(@
p
k
),
where R
l
jik
denotes the components of the curvature
tensor of r
Q
, turns out to be a suitable super-
derivation to start the Fedosov construction with
std
. In fact, the square of r
T
Q
þB turns out to
equal the square of r
T
Q
and all the other
preconditions of Fedosov’s construction are easily
verified just replacing r by r
T
Q
þB. The particular
property of the resulting star product
std
for =0
on T
Q is that it is a standard ordered star product,
that is, for all f 2C
1
(T
Q)[[]] and all 2
C
1
(Q)[[]] one has
std
f ¼
f ½41
and hence
std
in a certain sense is adapted to the
vertical polarization.
The methods mentioned so far can even be melted
into a more general situation, where one considers a
(complex) polarization on (M, !) and looks for star
products that are adapted to this polarization which
are then called polarized deformation quantizations
(cf. Donin (2003)). Here again a generalization of
Fedosov’s construction yields the existence and the
classification of such particular star products.
Another recent generalization of Fedosov’s con-
struction that goes beyond the framework of smooth
symplectic manifolds is that of the construction of
star products on symplectic orbispaces (cf. Pflaum
(2003)), which are stratified symplectic spaces. The
main idea there is to consider Fedosov’s construction
in local orbicharts and to show that the changes of
orbicharts induce isomorphisms between the locally
defined deformation quantizations, implying that the
locally defined products match together to define a
global deformation quantization on the symplectic
orbispace. To achieve this property, one has to adjust
the local Fedosov constructions appropriately, that is,
one has to use locally defined torsion-free symplectic
connections and formal series of closed 2-forms that
are related by the changes of the orbicharts.
Considering a vector bundle E !M, the sections
1
(E)arenaturallyaC
1
(M)-right module and a
1
(End(E))-left module, and it is a natural question
whether this bimodule structure can be deformed
such that
1
(E)[[]] becomes a (C
1
(M)[[]], ?)-right
module and a (
1
(End(E))[[]],w)-left module, where
w is a deformation of the usual composition of
elements of
1
(End(E)). In order to construct such
deformations, one can also adapt Fedosov’s construc-
tion (cf. Waldmann (2002)) considering W
E=(X
1
s= 0
1
(
W
s
T
M
V
T
ME))[[]] and W
End(E)=(X
1
s=0
1
(
W
s
T
M
V
T
MEnd(E)))[[]]
and extending the product to these spaces in
a natural way making WE a(W
End(E), )(W, )-bimodule. Furthermore, one
has to consider a connection r
E
that naturally induces
aconnectiononEnd(E), and both have to be added
to r to define the corresponding substitute of r on
the respective space. Then the Fedosov construction
with W End(E) can be considered yielding a
Fedosov derivation D
End(E)
with square zero, hence
a Fedosov–Taylor series
End(E)
and an associative
deformation F# G= (
End(E)
(F)
End(E)
(G)) of the
usual composition of sections in the endomorphism
bundle. Moreover, there is a map D
E
on WE
that is a superderivation with respect to the
bimodule multiplication along D
End(E)
and D ,
respectively. This map also has square zero and
the intersection of its kernel with the elements of
antisymmetric degree is in bijection to
1
(E)[[]] via
a natural generalization
E
of the Fedosov–Taylor
298 Fedosov Quantization
series. Defining F s:=(
End(E)
(F)
E
(s)) and s f :=
(
E
(s)(f )),
1
(E)[[]] can be given the structure
of a (
1
(End(E))[[]],#)-(C
1
(M)[[]], )-bimodule
which is indeed a deformation of the classical
bimodule structure of
1
(E). It is rather evident that
the same procedure also works for other products on
W and the above generalizations, in particular for
the product
Wick
on a Ka¨hler manifold, where one can
obtain (
1
(End(E))[[]],#
Wick
)-(C
1
(M)[[]],
Wick
)-
bimodules that are adapted to the complex structure in
case the curvature endomorphism of the connection
r
E
is of type (1, 1). For example, this holds true for
(anti-) holomorphic vector bundles endowed with a
Hermitian fiber metric h and the corresponding
connection that is compatible with h and the (anti-)
holomorphic structure.
Finally, the proof of existence of deformation
quantizations on arbitrary Poisson manifolds
(M, ), that includes a concrete construction
starting from Kontsevich’s star product on the
flat space R
n
equipped with a Poisson tensor,
given by Cattaneo et al. (2002) is similar in spirit
to Fedosov’s construction. There one constructs
two bundles J
1
and J
1
of associative algebras,
where – as a bundle – J
1
is isomorphic to J
1
[[]]
and J
1
is the bundle of infinite jets of smooth functions
on M which is equipped with the canonical flat
connection D
0
. The Poisson tensor gives rise to
the structure of a Poisson algebra on each fiber of J
1
and the canonical map C
1
(M) !J
1
yields a Poisson
algebra isomorphism between C
1
(M) and the
Poisson algebra of D
0
-flat sections in J
1
. The second
step in the construction consists in a deformation of
this correspondence. Using the Kontsevich formula for
R
n
,eachfiberofJ
1
can be equipped with an
associative product which is a deformation of the
above product on the fibers of J
1
in the direction of
the Poisson bracket induced by . Then analogously to
Fedosov’s construction, one constructs a compatible
connection D = D
0
þ D
1
þ
2
D
2
þ which is a
deformation of D
0
. Here compatibility just means
that D is a derivation with respect to the above
product on sections in J
1
implying that the D-flat
sections form a subalgebra. Moreover, one can
achieve that this connection is flat and in this case
C
1
(M)[[]] turns out to be in bijection to the D-flat
sections in J
1
. For the proof of existence of D and for
its recursive determination using an adaption of
Fedosov’s method, again special cases of Kontsevich’s
formality theorem prove to be the crucial tools. Pulling
back the above fiberwise product to C
1
(M)[[]] via
this isomorphism, one then obtains a star product on
(M, ). Since this isomorphism can be determined
recursively, the star product can in principle be
computed explicitly.
See also: Deformation Quantization; Deformation
Quantization and Representation Theory; Deformation
Theory; Deformations of the Poisson Bracket on a
Symplectic Manifold; String Field Theory.
Further Reading
Bayen F, Flato M, Frønsdal C, Lichnerowicz A, and Sternheimer
D (1978) Deformation theory and quantization. Annals of
Physics 111: 61–110 Part I.
Bayen F, Flato M, Frønsdal C, Lichnerowicz A, and Sternheimer D
(1978) Deformation theory and quantization. Annals of Physics
111: 111–151 Part II.
Bertelson M, Cahen M, and Gutt S (1997) Equivalence of star
products. Classical and Quantum Gravity 14: A93–A107.
Bordemann M and Waldmann S (1997) A Fedosov star product
of the Wick type for Kahler manifolds. Letters in Mathema-
tical Physics 41: 243–253.
Bordemann M, Neumaier N, and Waldmann S (1998) Homo-
geneous Fedosov star products on cotangent bundles. I. Weyl
and standard ordering with differential operator representa-
tion. Communications in Mathematical Physics 198: 363–396.
Cattaneo AS, Felder G, and Tomassini L (2002) From local to
global deformation quantization of Poisson manifolds. Duke
Mathematical Journal 115: 329–352.
Deligne P (1995) De´formations de l’Alge` bre des Fonctions d’une
Varie´te´ Symplectique: Comparaison entre Fedosov et
DeWilde, Lecomte. Selecta Mathematical, New Series 1(4):
667–697.
Donin J (2003) Classification of polarized deformation quantiza-
tions. Journal of Geometry and Physics 48: 546–579.
DeWilde M and Lecomte PBA (1983) Existence of star-products
and of formal deformations of the Poisson Lie algebra of
arbitrary symplectic manifolds. Letters in Mathematical
Physics 7: 487–496.
Fedosov BV (1985) Formal Quantization. In: Some Topics of
Modern Mathematics and Their Applications to Problems of
Mathematical Physics, Moscow, pp. 129–136.
Fedosov BV (1986) Quantization and the index. Soviet Physics–
Doklady 31(11): 877–878.
Fedosov BV (1994) A simple geometrical construction of
deformation quantization. Journal of Differential Geometry
40: 213–238.
Fedosov BV (1996) Deformation Quantization and Index Theory.
Berlin: Akademie Verlag.
Gutt S and Rawnsley J (1999) Equivalence of star products on a
symplectic manifold; an introduction to Deligne’s C
ˇ
ech
cohomology classes. Journal of Geometry and Physics 29:
347–392.
Karabegov AV (1996) Deformation quantizations with separation
of variables on a Ka¨ hler manifold. Communications in
Mathematical Physics 180: 745–755.
Karabegov A and Schlichenmaier M (2001) Almost-Ka¨hler
deformation quantization. Letters in Mathematical Physics
57: 135–148.
Kontsevich M (2003) Deformation quantization of Poisson
manifolds. Letters in Mathematical Physics 66: 157–216.
Mu¨ ller-Bahns MF and Neumaier N (2004) Some remarks on
g-invariant Fedosov star products and quantum momentum
mappings. Journal of Geometry and Physics 50: 257–272.
Nest R and Tsygan B (1995) Algebraic index theorem. Commu-
nications in Mathematical Physics 172: 223–262.
Neumaier N (2002) Local -Euler derivations and Deligne’s
characteristic class of Fedosov star products and star products
Fedosov Quantization 299
of special type. Communications in Mathematical Physics 230:
271–288.
Neumaier N (2003) Universality of Fedosov’s construction for
star products of Wick type on pseudo-Ka¨ hler manifolds.
Reports on Mathematical Physics 52: 43–80.
Omori H, Maeda Y, and Yoshioka A (1991) Weyl manifolds and
deformation quantization. Advances in Mathematics 85:
224–255.
Pflaum M (2003) On the deformation quantization of symplectic
orbispaces. Differential Geometry and Its Applications 19:
343–368.
Waldmann S (2002) Morita equivalence of Fedosov star products
and deformed Hermitian vector bundles. Letters in Mathema-
tical Physics 60: 157–170.
Feigenbaum Phenomenon see Universality and Renormalization
Fermionic Systems
V Mastropietro, Universita
`
di Roma ‘‘Tor Vergata’’,
Rome, Italy
ª 2006 Elsevier Ltd. All rights reserved.
Quantum Statistics
Quantum particles are described by a complex, square-
integrable wave function (x
1
, ..., x
N
)withjj
2
representing the probability density of finding N
particles at positions x
1
, x
2
, ..., x
N
, which will be
assumedtobeinad-dimensional square box V with
side L and periodic boundary conditions. If the N
particles are identical, jj
2
must be totally symmetric in
the exchange of any pair of coordinates. Regarding the
symmetry properties of itself, it is an experimental
fact (which finds its theoretical explanation in the
context of relativistic quantum field theory) that only
two possibilities can arise: either is symmetric or it
is antisymmetric, which means that (x
1
, ..., x
N
) =
(1)
P
(x
P
1
, ..., x
P
N
), where P
1
, ..., P
N
is a permuta-
tion of 1, ..., N,and(1)
P
is the parity of the
permutation. Particles described by a symmetric wave
function are called bosons, while particles with an
antisymmetric wave function are called fermions, after
Bose and Fermi, who introduced these concepts. The
fermionic wave function therefore vanishes if two
coordinates are equal, a property called Pauli exclu-
sion principle. Particles have an intrinsic quantized
angular momentum called spin and particles with
semi-integer spin are fermions, while particles with
integer spin are bosons. Examples of fermions are
electrons, protons, or neutrons, with spin = h=2,
where h is the Plank constant; examples of bosons are
phonons or mesons with integer spin.
The time evolution of a wave function is driven
(through the Schro¨ dinger equation) by the Hamilto-
nian operator, and the choice of such an operator is
determined by the physical system we want to
describe. One of the most important physical realiza-
tions of a fermionic system is given by the conduction
electrons in solids with a crystalline structure (like
metals). According to the classical theory of Drude, a
crystal can be described as a lattice of atoms in which
the valence electrons are lost by the atoms (which
become ions) and move freely in the metal; they are
responsible for the conduction properties of the
crystal. However, if one assumes that the electrons
are classical particles (in the sense that they obey the
Newtonian mechanics), one obtains wrong predictions
about the properties of crystals. One has to take into
account that the conduction electrons are quantum
particles and this provides us with a natural example
of a fermionic system; the Hamiltonian can be taken as
H
N
¼
X
N
i¼1
h
2
@
2
x
i
2m
þ uc x
i
ðÞ
"#
þ
X
i<j
v x
i
x
j
½1
The first term represents the nonrelativistic kinetic
energy of the electrons (m is the mass), uc(x)isa
periodic potential due to the ions in the lattice
(c(x) = c(x þ R) with R = (n
1
a
1
, ..., n
d
a
d
),n
i
2 Z)
and v(x y) is a two-body interaction potential,
which is modeled by a short-range potential to take
into account, phenomenologically, the electrostatic
screening. Finally, and u are couplings which
measure the ‘‘strength’’ of the corresponding inter-
action. Much more complicated and ‘‘realistic’’
Hamiltonians could be considered; for instance,
one can add an interaction with a stochastic field
to take into account impurities in the lattice, or with
a boson field to take into account the dynamics of
the ions, and so on. Note also that one can study not
only three-dimensional Fermi systems (d = 3), but
also d = 2ord = 1 systems; they can describe the
conduction electrons of crystals that are anisotropic
and should be considered as bidimensional or one-
dimensional systems. We focus on the nonrelativistic
300 Fermionic Systems
fermionic systems with Hamiltonian [1], which is a
problem of great importance from both the con-
ceptual and the applications poin t of view.
Second-Quantization Formalism
The Hilbert space of states of a system of N > 1
fermions is the space H
N
of all the complex square-
integrable antisymmetric functions (x
1
, ..., x
N
). Let
{
k
(x)}
k2R
d
be a basis for H
1
(the one-particle Hilbert
space of all the complex square-integrable functions
(x
1
)), where k is an index called quantum number.
Usually, the set of
k
(x) is chosen as the eigenfunctions
of the single-particle Hamiltonian
h
2
@
2
x
2m
þ ucðxÞ
for instance, if u = 0 then
k
ðxÞ¼
1
L
d=2
e
ikx
with hk representing the momentum; due to periodic
boundary conditions, k has the form k = (2=L)n,
n = n
1
, ..., n
d
with n
i
integer and [L=2] n
i
[(L 1)=2]. If we call jk
1
, ..., k
N
i the normalized
antisymmetrization of
k
1
(x
1
)
k
2
(x
2
)
k
N
(x
N
)
(Slater determinant), then the set of all possible
jk
1
, ..., k
N
i is a basis for H
N
; jk
1
, ..., k
N
i describes a
state in which the N fermions have quantum numbers
k
1
, ..., k
N
. One can introduce (Negele and Orland
1988, Berezin 1966) the creation or annihilation
operators a
þ
k
, a
k
: they are anticommuting operators,
fa
þ
k
; a
k
0
ga
þ
k
a
k
0
þ a
k
0
a
þ
k
¼
k;k
0
fa
þ
k
; a
þ
k
0
g¼fa
k
; a
k
0
g¼0
½2
such that a
þ
k
jk
1
, ..., k
N
i= jk, k
1
, ..., k
N
i if k 6¼ k
i
,
i = 1, ..., N and 0 otherwise; a
k
is the adjoint of
a
þ
k
. The action of a
þ
k
is to create a particle with
quantum number k if it is not present in the state,
and to yield zero otherwise (according to the Pauli
principle). The state j0i such that a
k
j0i= 0 for all k
is called the vacuum state and it represents a
state with no particles. The Fock space is defined as
the direct sum of the Hilbert spaces with any
number of particles, and all the elements of the
Fock space can be generated by linearly super-
posing products of creation operators acting over
the vacuum state. We can extend such definitions
by adding a label to such operators to take into
account the spin of the particle; for example, a
k,
are
creation or annihilation operators of a particle with
spin and position k.Intermsofa
þ
x,
= L
d=2
P
k
k
(x)a
þ
k,
and of its adjoint a
x,
, the Hamiltonian
can be written as
H ¼
X
Z
V
dx a
þ
x;
h
2
@
2
x
2m
a
x;
"
þu
Z
V
dx cðxÞa
þ
;x
a
x;
þ
X
;
0
Z
V
dx
Z
V
dy vðx yÞa
þ
x;
a
x;
a
þ
y;
0
a
y;
0
½3
According to the postulates of quantum statistical
mechanics, the grand canonical partition function is
given by Z = tr e
(HN)
,where = (T)
1
, is
the Boltzmann constant, T is the temperature, is the
chemical potential, N =
P
R
dx a
þ
x,
a
x,
, and tr is the
trace operation over the Fock space. The thermodyna-
mical average of an observable O is given by <O>=
Z
1
tr[e
(HN)
O]. Given a fermionic system, one is
often interested in its Schwinger functions defined as
follows: if x = (x,t)andt
1
t
2
t
s
, s even, then
Sðx
1
; x
2
; ...; x
s
Þ
¼
tr e
ðt
1
ÞðHNÞ
"
1
x
1
e
ðt
1
t
2
ÞðHNÞ
"
2
x
2
e
t
s
ðH NÞ
tr e
ðHNÞ
½4
with "
i
= , =2 t
i
=2; periodic and anti-
periodic boundary conditions are, respectively,
imposed over x
i
and t
i
. From the knowledge of the
Schwinger functions, one can compute all the
thermodynamical properties of a system at equili-
brium or close to equilibrium.
The Free Fermi Gas
Computation of the physical observables corre-
sponding to the complete Hamiltonian [3] is a
very difficult task. The natural starting point
consists in taking into account only the kinetic
term by putting = u = 0in[3], obtaining the free
Fermi gas model. The resulting model is not trivial
at all; its properties are radically different with
respect to the ones of a gas of classical particles,
and it is sufficient to understand many properties of
matter (see, e.g., Mahan (1990)). If
k
ðxÞ¼
1
L
d=2
e
ikx
then jk
1
,
1
, ..., k
N
,
N
i are eigenfunctions of
H with eigenvalue
P
k,
"(k)n
k,
, where "(k) =
h
2
jkj
2
=2m and n
k,
= 0,1, the occu pation number,
is the eigenvalue a
þ
k,
a
k,
; n
k,
= 1 if in the state there
is a fermion with mom entum k and spin , and it is
zero otherwise. The eigenfunction ji of H with
lowest energy is called the ground state, and it
determines the low-temperature properties of the
system. In order to find the ground state ji, one has
Fermionic Systems 301
to minimize
P
k,
"(k)n
k,
with the constraint that n
k
,
can take only the values 0 or 1 and
P
k,
n
k,
= N;if
there are many solutions to this problem, one says that
the ground state is degenerate. An approximate
solution is the following: if d = 3, one can consider a
state such that n
k;
= 1ifk is in a sphere of radius k
F
and zero otherwise; since the number of momenta
k = (2=L)n in the sphere is approximately given by
2
4k
2
F
3
L
3
8
2
we can choose k
F
= (3
2
)
1=3
, with = NL
3
.
The state
Q
jkjk
F
a
þ
1=2, k
a
þ
1=2, k
j0iis not the true ground
state when N,
L
are finite, but it is a very good
approximation of it and converges to it (in a suitable
sense) in the limit N,L !1, fixed. The boundary of
the sphere with radius k
F
in the space of momenta is
called the Fermi surface, and it is a key notion in the
theory of Fermi systems; if d = 2, it is replaced by a
circle and in d = 1bytwopoints.
Coming to the thermodynamical properties, the
partition function is given by
Z ¼
Y
k
X
n
k
¼0;1
e
ð"ðkÞÞn
k
¼
Y
k
ð1 þ e
ð"
k
Þ
Þ
and the specific heat by
C
v
¼
@
@T
@
@
log Z
One finds, by expressing in terms of through the
relation N = @f =@, that if d = 3, in the L !1
limit
C
v
¼
2
2
T
"
F
þ O
T
"
F
2
where "
F
=h
2
k
2
F
=2m. Early models for metals
described the electrons as classical particles; however
in such a case, a well-known result of classical
statistical mechanics states that they should contribute
to the specific heat by
3
2
, while experimentally their
contribution is much smaller. The solution of this
puzzle was provided by the above formula for C
v
;the
classical value is in fact depressed by a factor
2
3
T
"
F
which at room temperatures is O(10
2
), in agree-
ment with experimental data. The a verage number
of electrons with momentum hk is given, in the
infinite-volume limit, by
a
þ
k;
a
k;
DE
¼ð1 þ e
ð "ðkÞÞ
Þ
1
At zero temperature, it reduces to (jkjk
F
), that
is, it has a discontinuity at the Fermi surface, while
at high temperatures it is very close to the Maxwell
distribution ’e
("(k))
.
Finally, in the free Fermi gas model, all Schwinger
functions can be computed. One finds that, if, for
instance, "
i
= þ for i = 1, 2, ..., s=2 and "
i
=
otherwise, that the Schwinger function with s 4
can be expressed as sum of products of the s = 2
Schwinger function (also called the propagator)
Sðx
1
; ...; x
s
Þ¼
X
ð1Þ
Y
i;j
S
0
x
i
x
ðjÞ
½5
where i = 1, ..., s=2, j = s=2 þ 1, ..., s,
j
is a per-
mutation of j = s=2 þ 1, ..., s,(1)
is the parity of
this permutation,
P
is the sum over all the
possible permutations; such a formula is called the
Wick rule. By an explicit computation, S
0
(x y)is
given by
2
X
k
0
¼2ðn
0
þ1=2Þ
1
2
L
d
X
k¼ð2=LÞn
e
ikðxyÞ
ik
0
þjkj
2
=2m
2
X
k
0
¼2ðn
0
þ1=2Þ
1
2
L
d
X
k¼ð2=LÞn
e
ikðxyÞ
^
S
0
ðkÞ
½6
where k = (k
0
, k). In the limit L, !1, for large
distances S(x, y) decays as a power law, O(jx yj
1
)
times an oscillating function of period k
1
F
. Note that
S
0
(k) in the limit , L !1diverges for k
0
= 0 and
"(k) = , that is, at the Fermi surface ( = "
F
in the
limit !1); when is finite, S
0
(k) is finite even for
L !1, that is, the finite temperature acts as an
infrared cutoff.
Fermions in an External Potential
The next step consists in adding an external periodic
potential to the free Fermi gas model, taking into
account the field generated by the ions of the lat tice.
We consider then [3] with = 0 and u 6¼ 0. As in the
previous case, the eigenfunctions of the N-particle
Hamiltonian can be computed and are expressed in
terms of the single-particle eigenfunctions of
h
2
@
2
x
=2m þ uc(x); they are called Bloch waves
and have the form
k
ðxÞ¼
1
ffiffiffiffiffiffiffiffiffi
L
d=2
p
e
ikx
u
k
ðxÞ; u
k
ðxÞ¼u
k
ðx þ RÞ
k, called the crystalline momentum, is conserved
modulo G, the vectors of the reciprocal lattice,
defined as
G ¼ 2
n
1
a
1
; ...;
n
d
a
d
302 Fermionic Systems
The eigenvalue "(k)ofh
2
@
2
x
=2m þ uc(x) associated
with a Bloch wave
k
(x) has some peculiar properties;
in the L !1 limit, one finds that "(k) is not a
continuous function (unlike the u = 0 case) but it has
gaps, that is, first-order discontinuities. For d = 1,
by a convergent power-series expansion in u, one
finds that "(k) is a continuous monotonically increas-
ing function except at the points n=a, n an integer;
at these points "(R) is discontinuous and "((n=a)
þ
)
"((n=a)
)
n
= u
^
c
n
þ O(u
2
); the gaps divide "(k)
into disconnected pieces called energy bands. Some-
thing similar happens in d = 2, 3, in which gaps open
for R such that G
2
þ 2kG = 0.
Again, the eigenfunctions of H are given by
jk
1
,
1
, ..., k
N
,
N
i with eigenvalue
P
k,
"(k)n
k,
,
and the Fermi surface is still defined by the set k
such that "(k) = "
F
with "
F
determined by the
condition
P
k:"(k)
"
F
1 = N. However, in this case
the Fermi surface is not anymore a sphere in d = 3,
but it is in general a polyhedron of a very complex
shape. The Schwinger functions are expressed by the
Wick rule [5] in terms of the two-point Schwinger
functions; they are given by [6] with e
ik(xy)
replaced
by
k
(x)
k
(y) and jk j
2
=2m replaced by "(k). The
asymptotic properties of the two-point Schwinger
function are quite different with respect to the u = 0
case. This is easy to see if d = 1; in the limit L, !1,
S(k) is singular if does not belong to the interval
["((n=a)
þ
), "((n=a)
)], whereas it is finite if
belongs to such an interval; in the first case, S(x, y)
decays for large distances as O(jx yj
1
), whereas in
the second case it is O(e
j
n
jjxyj
). This means that,
depending on the number of particles (which essen-
tially fixes ), the Schwinger function has a totally
different asymptotic behavior. This fact has impor-
tant consequences in many physical properties; for
instance, the conductivity (which can be computed
from the s = 4 Schwinger function) vanishes if
belongs to the interval ["((n=a )
þ
), "((n=a)
)]. Simi-
lar properties hold for d = 2, 3; hence, from the
knowledge of the number of particles and the periodic
potential generated by the ions, one can predict if the
system is an insulator or a metal.
Note also that the conductivity is infinite in the
infinite-volume and zero-temperature limit, when
does not correspond to a gap; in other words, the
electric current in a perfect crystal lattice is not
subjected to any dissipation of energy. A finite
resistivity is found only if one takes into account
deviations from perfect periodicity. To simulate
impurities in the lattice, one can add, according to
Anderson, to the Hamiltonian an interaction term
of the form
x
þ
x
x
, where
x
is a Gaussian
stochastic field. A detailed mathematical investi-
gation has been devoted to the properties of
eigenfunctions of h
2
@
2
x
=2m þ
x
, where
x
is a
Gaussian field (see, e.g., Pastur and Figotin (1991));
it is found that if is large enough in d = 2, 3 and
for any in d = 1, the single-particle eigenfunctions
are exponentially localized, that is, they decay
exponentially at large dist ances; this implies a finite
conductivity. One can also add to the Hamiltonian a
term
0
x
þ
x
x
, with
0
x
a quasiperiodic function, in
order to describe crystals in which the lattice
develops a periodic distortion, with incommensurate
period with respect to the lattice periodicity. For
d = 1 and large, one again finds localized
eigenfunctions, whereas for small there are
extended states (see, e.g., Pastur and Figotin
(1991)); such results are obtained with the
Kolgomorov–Arnol’d–Moser (KAM) techniques.
Interacting Systems
The analysis of noninteracting Fermi systems has
been very successful in understanding qualitatively
many features of crystals, but there are many
properties (e.g., superconductivity or magnetism)
which cannot be really explained without taking
into account the interaction between fermions;
however, the analysis becomes more involved.
When there is no interaction, the properties of
the many-body system can be understood in terms of
the single-body properties; the eigenfunctions of the
Hamiltonian are, in fact, obtained in terms of the
single-particle eigenfunctions. This is not true when
6¼ 0 when a description of the system in terms of
independent particles is impossible. In order to
compute the interacting Schwinger functions, it is
convenient to write them in terms of fermionic
functional integrals (Berezin 1966). One introduces
a set of anticommuting Grassmann variables
þ
k
,
k
,
k = (k
0
, k); the Grassmann integration is defined by
R
d
k
k
= 1and
R
d
k
= 0, = , and the integral
of any analytic function of the Grassmann variables
can be obtained by expanding it in Taylor series
(which is a finite sum if suitable cutoffs are imposed
and L, are finite) and using the above rules; finally,
þ
x
¼
1
L
d
X
k
k
ðxÞe
ik
0
t
þ
x
and
x
is defined in an analogous way. The
Schwinger function can be written as a Grassmann
integral as follows:
Sðx
1
; x
2
; ...; x
N
Þ
¼
@
N
@
"
i
x
1
...@
"
i
x
N
log
Z
Pðd Þe
þ
R
dx
"
x
"
x
j
’¼0
½7
Fermionic Systems 303
where P(d ) is the fermionic integration
[
Q
k
d
þ
k
d
k
] exp [
P
k
þ
k
[ik
0
þ"(k) ]
k
], while
y = (y, s), and
¼
X
;
0
Z
dx dy vðx yÞ
ðt sÞ
þ
x;
x;
þ
y;
0
y;
0
½8
The Grassmann integral of a monomial of Grassmann
variables can be obtained by the Wick rule [5] with
propagator the Fourier transform of ( ik
0
þ "(k)
)
1
. As stated earlier, the propagator is finite at
nonzero temperature, whereas if = 1,thenitis
singular when k = (k
0
, k)issuchthatk
0
= 0and
"(k) = .
One can write [7] as a series by Taylor-expanding
the exponential and using the Wick rule; each order
of the expansion can be represented as a sum of
Feynman diagrams, very similar to the ones appear-
ing in quantum field theory. We have then an
algorithm to compute [7]; nevertheless, to extract
information from such a series is quite difficult. One
cannot really compute an infinite (in the L = 1
limit) number of coefficients, so one is tempted, for
small , to compute only the first few of them,
neglecting the others. However, it appears that this
approximation is generally not justified, and it leads
to wrong resul ts; the reason is that the Schwinger
functions for = 0or 6¼ 0 are not analytically
close, or, in more physical terms, even if is small,
the physical behavior of the free and interacting
theories can be quite different, especially at low
temperatures. A number of very interesting concepts
(e.g., spontaneous symmetry breaking or the mass
generation phenomenon) , or techniques (e.g., the
renormalization group method, or the parquet or
random phase approximation) have been introduced
in the last 50 years to analyze [7], and indeed many
results have been obtained which explain several
physical propert ies of the matter, such as super-
conductivity or the Kondo eff ect (see, e.g., Anderson
1985, Abrikosov et al. 1965, Mahan 1990, Negele
and Orland 1988, Pines 1961). Unfortunately, most
of such results are not really mathematically
consistent, and in many cases quantitative computa-
tions are impossible (in computations one generally
neglects terms which, according to a heuristic
physical intuition, are irrelevant, but no control of
the error introduced by this approximation is
attempted). In recent times, attempts towards a
mathematical understanding of the functional inte-
gral [7] have started (see, e.g., Benfatto and
Gallavotti (1995), and references therein); the
methods rely on the mathematical implementation
of Wilson’s renormalization group methods via
multiscale analysis (Gallavotti 1985). The neces sity
of a firmer mathematical basis was felt mainly
under the pressure of the recent discovery of high-
T
c
superconductors whose behavior is still not
understood in terms of the microsopic model [7];
this has forced reconsideration of the validity of the
approximations usually made in the analysis of this
model.
The behavior of [7] depends crucially on the
temperature. At high temperatures, we can simply
expand the expo nential in [7] in a power series of ,
and find that each Feynman graph contributing to
the nth perturbative order is bounded by C
n
jj
n
,
with C
C
for some constants C, ; this follows
immediately by using the Wick rule and by
remembering that the propagator is larger than
O(
1
). As the number of Feynman graphs con-
tributing to order n is O(n!), a bound on each
Feynman graph is not sufficient to prove the
convergence of the series. To prove convergence,
one has to take into account cancellations, due to
the anticommutativity of fermionic variables. Such
cancellations are proved via Gram’s inequality for
determinants and a bound C
n
n
can be obtained for
the order n (without factorials); hence, convergence
follows for temperatures greater than O(jj
) for
some constant >0. One finds that
S(k) = S
0
(k)(1 þ A
(k)) with jA
(k)jCjj, that is,
the interaction has essentially no influence on the
physical properties of the system at high
temperatures.
Landau Fermi Liquids
We consider next an intermediate region of tem-
peratures, that is, e
a=jj
T jj
for some con-
stants a, . In this region, the naive expansion in
power series of fails and other techniques, such as
renormalization group, are necessary. Such a
method allows us to perform a suitable resummation
of the naive power series in , and one gets, for
small enough, T e
a=jj
and "(k) = jkj
2
=2m,
^
SðkÞ¼
1
ZðÞ
1 þ A
ðkÞ
ik
0
þ v
F
ðÞ½jkjk
F
ðÞ
½9
where Z() = 1 þ z(), v
F
() =hk
F
=m þ v(), and
k
F
() = k
F
þ (), with z() = O(
2
), () = O(),
v() = O(
2
), and z(), (), v
F
() essentially tem-
perature independent; moreover, jA
(k)j is O().
The above formula has been proved rigorously for
d = 2(seeRivasseou (1994), and references therein);
for d = 3, it has been proved at the level of formal
perturbation theory (Benfatto and Gallavotti 1995).
The case "(k) = jkj
2
=2m is quite special, as the shape
304 Fermionic Systems
of the interacting Fermi surface is fixed by the
rotation-invariant symmetry; it is necessarily circular
(d = 2) or spherical (d = 3), whereas in general the
interaction can also modify its shape. For d = 2, if the
interacting Fermi surface is symmetric, smooth and
convex, a formula like [9] still holds (with a function
k
F
(, k)replacingk
F
()) up to exponentially
small temperatures (see references in Gentile and
Mastropietro (2001)).
It is apparent from [9] that one cannot derive such
a formula from a power-series expansion in ;by
expanding [9] as a series in , one immediately finds
that the nth term is O(
n
n
), which means that the
naive perturbative expansion cannot be convergent
up to exponentially small tem peratures. It can be
derived only by selecting and resumming some
special class of terms in the original expansion. A
peculiar property of [9] is that the wave function
renormalization Z() is essentially independent of
the temperature. Such temperature independence is a
consequence of cancellations in the perturbative
series essentially due to the curvature of the Fermi
surface. For d = 1, a formula similar to [9] is also
valid; however, such cancellations are not present
and one finds Z() = 1 þ O(
2
log ). Comparing
S(k) given by [9] with the Fourier transform S
0
(k)of
[6], we note that the Schwinger function of the
interacting system is still very similar to the
Schwinger function of a free Fermi gas, with
physical parameters (e.g., the Fermi momentum,
the wave function renormalization, or the Fermi
velocity) which are changed by the interaction. This
property is quite remarkable: the eigenstates cannot
be constructed when = 0 starting from the single-
particle states but, nevertheless, the physical proper-
ties of the interacting system (which can be deduced
from the Schwinger functions) are qualitat ively very
similar to the ones of the free Fermi gas, although
with different parameters; this explains why the free
Fermi gas model works so well to explain the
properties of crystals, although one neglects the
interactions between fermions which are, of course,
quite relevant. A fermionic system with such a
property is called a Landau Fermi liquid (see, e.g.,
Arbikosov et al. 1965, Mahan 1990, Pines 1961),
after Landau, who postulated in the 1950s that
interacting systems may evolve continuously from
the free system in many cases.
It was generally accepted that metals in this range
of temperatures were all Landau Fermi liquids
(except one-dimensional systems). However, the
experimental discovery of the high-T
c
superconduc-
tors (see, e.g., Anderson (1997)) has changed this
belief, as such metals in their normal state, that is,
above T
c
are not Landau Fermi liquids; their
wave function renormalization behaves like 1 þ
O(
2
log ) instead of 1 þ O(
2
) as in Landau
Fermi liquid. This behavior has been called
marginal-Fermi-liquid behavior and many attempts
have been devoted to predict such behavior from [7].
In order to see deviations from Fermi liquid
behavior, one could consider Fermi surfaces with
flat or almost flat sides or corners (which are quite
possible; e.g., in a square lattice with one conduc-
tion electron per atom, such as in the ‘‘half-filled
Hubbard model’’).
Let us finally consider the last regime, that is,
temperatures lower than O(e
a=jj
). Except for very
exceptional cases (e.g., asymmetric Fermi surfaces,
i.e., such that "(k) 6¼ "( k) except for a finite
number of points, in which Fermi liquid behavior
is found down to T = 0 (Feldman et al. 2002)), a
strong deviati on from Fermi liquid behavior is
observed; the interacting Schwinger function is not
similar to the free one and the physical properties in
this regime are totally new.
One-Dimensional Systems up to T = 0
The only case in which the Schwinger functions of
the Hamiltonian [3] can be really computed down to
T = 0 occurs for d = 1; in such a case, an expression
like [9] is not valid anymore and the system is not a
Fermi liquid. On the contrary, when u = 0 and for
small repulsive >0, one can prove, for spinning
fermions (see Benfatto and Gallavotti (1995), Gentile
and Mastropietro (2001) and references therein) that
^
SðkÞ¼
k
2
0
þ v
2
F
ðÞjkjk
F
ðÞðÞ
2
hi
ðÞ
ik
0
þ v
F
ðÞjkjk
F
ðÞ½
1 þ A
ðkÞ½
½10
where k
F
() = k
F
þ O() and () = a
2
þ O(
3
)isa
critical index. This means that the interaction
changes qualitatively the nature of the singularity
at the Fermi surface; S (k) is still diverging at the
Fermi surface but with an exponent which is no
longer 1 but is 1 2(), with () a nonuniversal
(i.e., -dependent) critical index. As a consequence,
the physical properties are different with respect to
the free Fermi gas; for instance, the occupation
number n
k
is not discontinuous at k = k
F
()when
T = 0. Nonuniversal critical indices appear in all
the other response functions. Fermionic systems
behaving in this way are called Luttinger liquids,
as they behave like the exactly solvable Luttinger
model describing relativistic spinless fermions
with linear dispersion relation. The solvability of
this model, due to Mattis and Lieb (1966), relies
Fermionic Systems 305