  We discuss the restricted linear group in infinite dimensions modeled by the
Schatten class of rank $2p=4$ which contains $(3+1)$-dimensional analog of the
loop groups and is closely related to Yang-Mills theory with fermions in
$(3+1)$-dimensions. We give an alternative to the construction of the ``highest
weight'' representation of this group found by Mickelsson and Rajeev. Our
approach is close to quantum field theory, with the elements of this group
regarded as Bogoliubov transformations for fermions in an external Yang-Mills
field. Though these cannot be unitarily implemented in the physically relevant
representation of the fermion field algebra, we argue that they can be
implemented by sesquilinear forms, and that there is a (regularized) product of
forms providing an appropriate group structure. On the Lie algebra level, this
gives an explicit, non-perturbative construction of fermion current algebras in
$(3+1)$ space-time dimensions which explicitly shows that the ``wave function
renormalization'' required for a consistent definition of the currents and
their Lie bracket naturally leads to the Schwinger term identical with the
Mickelsson-Rajeev cocycle. Though the explicit form of the Schwinger term is
given only for the case $p=2$, our arguments apply also to the restricted
linear groups modeled by Schatten classes of rank $2p=6,8,\ldots$ corresponding
to current algebras in $(d+1)$- dimensions, $d=5,7,\ldots$.
