  A natural way to generalise tensor network variational classes to quantum
field systems is via a continuous tensor contraction. This approach is first
illustrated for the class of quantum field states known as continuous
matrix-product states (cMPS). As a simple example of the path-integral
representation we show that the state of a dynamically evolving quantum field
admits a natural representation as a cMPS. A completeness argument is also
provided that shows that all states in Fock space admit a cMPS representation
when the number of variational parameters tends to infinity. Beyond this, we
obtain a well-behaved field limit of projected entangled pair states (PEPS) in
two dimensions that provide an abstract class of quantum field states with
natural symmetries. We demonstrate how symmetries of the physical field state
are encoded within the dynamics of an auxiliary field system of one dimension
less. In particular, the imposition of Euclidean symmetries on the physical
system requires that the auxiliary system involved in the class' definition
must be Lorentz-invariant. The physical field states automatically inherit
entropy area laws from the PEPS class, and are fully described by the
dissipative dynamics of a lower dimensional virtual field system. Our results
lie at the intersection many-body physics, quantum field theory and quantum
information theory, and facilitate future exchanges of ideas and insights
between these disciplines.
