In this article, we are concerned with “finite dimensional fermions,” by which we mean vectors in a finite dimensional complex space embedded in the exterior algebra over itself. These fermions are spinless but possess the characterizing anticommutativity property. We associate invariant complex vector fields on the Lie group Spin(2n + 1) to the fermionic creation and annihilation operators. These vector fields are elements of the complexification of the regular representation of the Lie algebra so(2n+1). As such, they do not satisfy the canonical anticommutation relations; however, once they have been projected onto an appropriate subspace of L2(Spin(2n + 1)), these relations are satisfied. We define a free time evolution of this system of fermions in terms of a symmetric positive-definite quadratic form in the creation–annihilation operators. The realization of fermionic creation and annihilation operators brought by the (invariant) vector fields allows us to interpret this time evolution in terms of a positive self-adjoint operator that is the sum of a second order operator, which generates a stochastic diffusion process, and a first order complex operator, which strongly commutes with the second order operator. A probabilistic interpretation is given in terms of a Feynman–Kac-like formula with respect to the diffusion process associated with the second order operator.

1.
Applebaum
,
D.
,
Lévy Processes and Stochastic Calculus
(
Cambridge University Press
,
2004
).
2.
Barut
,
A. O.
,
Božić
,
M.
, and
Marić
,
Z.
, “
The magnetic top as a model of quantum spin
,”
Ann. Phys.
214
(
1
),
53
83
(
1992
).
3.
Barut
,
A.
and
Raczka
,
R.
,
Theory of Group Representations and Applications
(
World Scientific Publishing Co., Inc.
,
1986
).
4.
Berezin
,
F. A.
,
The Method of Second Quantization
(
Academic Press
,
1966
).
5.
Bopp
,
F.
and
Haag
,
R.
, “
Über die möglichkeit von spinmodellen
,”
Z. Naturforsch. A
5
(
12
),
644
653
(
1950
).
6.
Combe
,
P.
,
Høegh-Krohn
,
R.
,
Rodriguez
,
R.
,
Sirugue
,
M.
, and
Sirugue-Collin
,
M.
, “
Poisson processes on groups and Feynman path integrals
,”
Commun. Math. Phys.
77
(
3
),
269
288
(
1980
).
7.
De Angelis
,
G. F.
,
Jona-Lasinio
,
G.
, and
Sidoravicius
,
V.
, “
Berezin integrals and Poisson processes
,”
J. Phys. A: Math. Gen.
31
(
1
),
289
(
1998
).
8.
Fukutome
,
H.
, “
On the SO(2N + 1) regular representation of operators and wave functions of fermion many-body systems
,”
Prog. Theor. Phys.
58
(
6
),
1692
1708
(
1977
).
9.
Fukutome
,
H.
,
Yamamura
,
M.
, and
Nishiyama
,
S.
, “
A new fermion many-body theory based on the SO(2N + 1) Lie algebra of the fermion operators
,”
Prog. Theor. Phys.
57
(
5
),
1554
1571
(
1977
).
10.
Fulton
,
W.
and
Harris
,
J.
,
Representation Theory: A First Course
(
Springer Science & Business Media
,
1991
).
11.
Gelfand
,
I. M.
and
Minlos
,
R. A.
,
Representations of the Rotation and Lorentz Groups and Their Applications
(
Martino Publishing
,
2012
).
12.
Glimm
,
J.
and
Jaffe
,
A.
,
Quantum Physics: A Functional Integral Point of View
(
Springer Science & Business Media
,
1987
).
13.
Goodman
,
R.
and
Wallach
,
N. R.
,
Symmetry, Representations, and Invariants
(
Springer Science & Business Media
,
2009
).
14.
Hall
,
B.
,
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
(
Springer
,
2015
), Vol. 222.
15.
Helgason
,
S.
,
Differential Geometry, Lie Groups, and Symmetric Spaces
(
Academic Press
,
1979
).
16.
Hida
,
T.
,
Kuo
,
H.-H.
,
Potthoff
,
J.
, and
Streit
,
L.
,
White Noise: An Infinite Dimensional Calculus
(
Springer Science+Business Media
,
Dordrecht
,
1993
).
17.
Ikeda
,
N.
and
Watanabe
,
S.
,
Stochastic Differential Equations and Diffusion Processes
(
North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo
1989
).
18.
Jørgensen
,
P. E. T.
, “
Representations of differential operators on a Lie group
,”
J. Funct. Anal.
20
(
2
),
105
135
(
1975
).
19.
Nelson
,
E.
, “
Analytic vectors
,”
Ann. Math.
70
(
3
),
572
615
(
1959
).
20.
Reed
,
M.
and
Simon
,
B.
,
I: Functional Analysis
(
Academic Press
,
1981
), Vol. 1.
21.
Rogers
,
L. C. G.
and
Williams
,
D.
,
Diffusions, Markov Processes and Martingales: Volume
2, Itô Calculus (
Cambridge University Press
,
1994
).
22.
Rosen
,
N.
, “
Particle spin and rotation
,”
Phys. Rev.
82
(
5
),
621
(
1951
).
23.
Schmüdgen
,
K.
,
Unbounded Operator Algebras and Representation Theory
(
Birkhäuser
,
1990
).
24.
Schulman
,
L. S.
,
Techniques and Applications of Path Integration
(
Dover Publications, Inc.
,
Mineola, NY
,
2005
).
25.
Schulman
,
L.
, “
A path integral for spin
,”
Phys. Rev.
176
(
5
),
1558
(
1968
).
26.
Simon
,
B.
,
The P(ϕ)
2Euclidean (Quantum) Field Theory (
Princeton University Press
,
1974
).
27.
Simon
,
B.
,
Functional Integration and Quantum Physics
(
American Mathematical Society
,
2005
).
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