The energy-momentum tensor (EMT) is the Noether current associated with translations. It is of interest because, first of all, it has physical meaning as it contains the energy density and the momentum density. Moreover, its trace can be related to the beta function so that the scaling behaviour of the theory at hand can be studied. We are particularly interested in the scaling behaviour of strongly coupled theories. To explore the strong coupling regime it is necessary to compute the EMT non-perturbatively, i.e. on the lattice. This complicates matters greatly. On the lattice translation invariance is broken which leads to additional terms in the translation Ward identity from which the EMT is derived. This results in turn in the need to renormalise the EMT on the lattice. In this thesis we extend recent studies on the renormalisation of the EMT in four-dimensional gauge theory to the case of a three-dimensional scalar theory to investigate its divergence structure and the numerical feasibility of the suggested procedure on a more basic level. Furthermore, scalar φ4-theory in three dimensions exhibits an infrared fixed point and can thus serve as a toy model to examine mechanisms for building theories beyond the standard model. Our strategy to renormalise the EMT on the lattice is to identify all possible terms that can mix with both sides of the translation Ward identity. The renormalised EMT is a combination of operators of the same or lower dimension obeying the symmetries of the theory. The mixing is determined by requiring that the renormalised EMT satisfies the correct Ward identities. Using different probes in the translation Ward identity one can compute the coefficients of the EMT by solving a linear system of equations. However, contact terms can arise. One solution is the recently introduced Wilson flow. Its renormalisation properties allow for expectation values free of contact terms. That way the Wilson flow provides for a meaningful theoretical formulation of the EMT on the lattice that can be used in practice. In this thesis we review the renormalisation properties and the phase diagram of scalar φ4-theory in three dimensions, the translation Ward identity and the EMT in the continuum, as well as the gradient flow for scalar theory. A large part is dedicated to the perturbative renormalisation of the EMT on the lattice. Finally, our strategy to compute the renormalisation constants of the EMT in scalar theory non-perturbatively is discussed in detail, and our results for the renormalisation constants are presented.

Supersymmetrische Feldtheorien können durch eine nicht-lineare und nicht-lokale Transformation der bosonischen Felder charakterisiert werden, der Nicolai-Abbildung. Sie bildet das wechselwirkende Funktionalmaß so auf das Maß der zugehörigen freien Theorie ab, dass die Jacobi-Determinante gleich dem Produkt der fermionischen Determinanten ist. Wir untersuchen die Nicolai-Abbildungen des 2-dimenionalen Wess-Zumino Modells und der N=1 sowie N=4 super Yang-Mills Theorien. Wir geben einen konstruktiven Beweis für die Existenz der Nicolai-Abbildung in diesen Theorien. Der Beweis beinhaltet die Herleitung des infinitesimalen Generators der inversen Abbildung, dem R-Operator. Wir benutzen diesen Operator, um die Nicolai-Abbildung im 2-dimensionalen Wess-Zumino Modell bis zur fünften Ordnung in der Kopplung zu berechnen. In der N=1 super Yang-Mills Theorie führen wir die Begriffe der on- bzw. off-shell Nicolai-Abbildungen ein. Die on-shell Abbildung existiert in d=3,4,6 und 10 Dimensionen aber nur für die Landau Eichung. Wir bestimmen sie bis zur vierten Ordnung. Die off-shell Abbildung existiert nur in d=4 Dimensionen aber für allgemeine Eichungen. Wir berechnen sie in der axialen Eichung bis zur zweiten Ordnung. Für die N=4 super Yang-Mills Theorie geben wir den R-Operator an und zeigen, dass man die N=4 Nicolai-Abbildung durch dimensionale Reduktion aus der N=1 Abbildung erhält. Inverse Nicolai-Abbildungen bilden Quantenkorrelationsfunktionen bosonischer Observablen auf freie Korrelationsfunktionen ab. Daher gestatten sie eine Quantisierung supersymmetrischer Theorien ohne die Verwendung von Fermionen oder Geistern. Wir benutzten diese Eigenschaft, um den Vakuum-Erwartungswert der Maldacena-Wilson Geraden bis zur sechsten Ordnung zu berechnen. Im zweiten Teil dieser Arbeit bestimmen wir die explizite Form aller Felder im 1/2-BPS Energie-Impuls-Tensor Multiplet in N=4 super Yang-Mills. Dieses Multiplet enthält den R-Symmetrie Fluss und den Energie-Impuls-Tensor. / Supersymmetric field theories can be characterized by the existence of a non-linear and non-local transformation of the bosonic fields, the Nicolai map. It maps the interacting functional measure to that of a free theory such that the Jacobian determinant of the transformation equals the product of the fermionic determinants. In this thesis, we study the Nicolai maps of the 2-dimensional Wess-Zumino model, N=1 super Yang-Mills and N=4 super Yang-Mills. We give a constructive proof for the existence of the Nicolai map in these theories. The proof includes the derivation of the infinitesimal generator of the inverse Nicolai map, called the R-operator. We use this operator to compute the Nicolai map of the 2-dimensional Wess-Zumino model up to the fifth order in the coupling. In N=1 super Yang-Mills, we introduce the notion of on- and off-shell Nicolai maps. The on-shell Nicolai map of N=1 super Yang-Mills exists in d=3,4,6 and 10 dimensions but is constrained to the Landau gauge. We compute this map up to the fourth order. The off-shell Nicolai map exists only in d=4 dimensions but for general gauges. We compute it in the axial gauge up to the second order. In N=4 super Yang-Mills, we give the R-operator and show that the Nicolai map can be obtained from the Nicolai map of 10-dimensional N=1 super Yang-Mills by dimensional reduction. Inverse Nicolai maps map quantum correlation functions of bosonic observables to free correlation functions. Hence, Nicolai maps allow for a fermion (and ghost) free quantization of supersymmetric (gauge) theories. We apply this property to compute the vacuum expectation value of the infinite straight line Maldacena-Wilson loop in N=4 super Yang-Mills to the sixth order. Thus extending the previous perturbative result by one order. In the second part of this thesis, we derive the explicit field content of the 1/2-BPS stress tensor multiplet in N=4 super Yang-Mills, which contains the R-symmetry current and the energy-momentum tensor.