MSci (Hons) Theoretical Physics with Mathematics Integrated Master's degree at Lancaster University

Explore electricity, magnetism, thermodynamics, and quantum physics, providing a strong foundation in classical and modern physics. You will learn about electric and magnetic fields and forces through developing an understanding of Maxwell’s equations and their application. You will study topics in thermodynamics including heat transfer and ideal gases. You will be introduced to quantum mechanics, by examining atomic models, wave-particle duality, and the Schrödinger equation. Through problem-solving and conceptual understanding, you will develop analytical skills applicable in physics, engineering, and research. You will learn to apply mathematical models, understand physical principles, an

At university, emphasis is placed on understanding general mathematical theorems. They apply in many different cases, and understanding why a result is true enables us to creatively use the underlying ideas to tackle new problems. Study the language and structure of mathematical proofs, illustrated by results from number theory. You will see the concept of congruence of integers, which is a simplified form of arithmetic where seemingly impossible problems become solvable. In relation, you’ll encounter the abstract idea of an equivalence relation. Sets and functions form the basic language of mathematics. You will study functions of a real variable and abstract functions between arbitrary set

Interested in how mathematicians build theories from basic concepts to complex ideas, like eigenvalues and integration? Journey from polynomial operations to matrices and calculus through this module. Starting with polynomials and mathematical induction, you will learn fundamental proof techniques. You will explore matrices, arrays of numbers encoding simultaneous linear equations, and their geometric transformations, which are essential in linear algebra. Eigenvalues and eigenvectors, which characterise these transformations, will be introduced, highlighting their role in applications including population growth and Google's page rankings. Next, we will reintroduce you to calculus, from its

Modern artificial intelligence relies on multivariate calculus: every time a neural network learns, it does so by computing derivatives in high-dimensional spaces. Many real-world problems seek to understand the function of a vector, where the vector could be a position in space, a direction, or the weights of a neural network. In this module, you will explore the world of multivariate techniques and multivariate calculus, deepening your understanding of vectors, angles, curves, surfaces and volumes, multidimensional space, and alternative co-ordinate systems. You will encounter multidimensional derivatives, integrals and stationary points, and practice multidimensional analogues of techniqu

Symmetry is central to our understanding of a range of subjects, from the structure of molecules to the roots of polynomials. In this module, you will see how group theory naturally appears whenever we look at symmetry. Using familiar examples, including symmetries of regular polygons, rotations and reflection matrices, roots of 1 in the complex plane, and permutations, you will define what makes a group and how this can provide a unifying language, highlighting connections between seemingly different subjects. You will then transition into mathematical analysis, developing an approach to sequences, limits, and continuity that provides the foundation for calculus. Examining a range of exampl

We introduce you to the fundamental nature of Physics and teach you key skills in the use of experiment and uncertainty, units, and dimensional analysis. You will study topics such as Newton’s laws of motion, rotation of rigid bodies and the gravitational force. You will also be introduced to more advanced concepts such as special relativity and Lagrangian mechanics. You will apply some of these concepts to astronomical problems such as determining escape speed, and the motion of satellites and planetary orbits. You will learn about some exotic phenomena like black holes and dark matter.

In this module you will explore electromagnetism through the beauty of Maxwell’s equations and the mathematical tools of vector calculus. Using these, you will be able to describe electromagnetic fields and waves created by simple configurations of charges and currents, and to model the effects of media on the propagation of electromagnetic waves. You will investigate the basic properties of wave propagation, diffraction and interference, and of simple optical instruments. This will enable you to make connections between the many different phenomena in nature that share the mathematical model of a harmonic oscillator or of a wave.

This module introduces experimental techniques and provides an overview of theoretical methods used in classical mechanics. You will undertake a short series of laboratory experiments where you will learn to assess errors and uncertainties. These sessions also introduce scientific instruments, measurement techniques and log-keeping skills. You will learn variational approaches to derive equations of motion within the frameworks of Lagrangian and Hamiltonian mechanics. This will allow you to exploit the power of these techniques by using appropriate generalised coordinates. You will also apply these methods to dynamical problems in classical mechanics, introducing the concepts of phase space,

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning. You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem. Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving

Learn how physics principles, including quantum mechanics and many-particle statistics, underpin the properties of materials. This will allow you to relate features of atoms, electrons and phonons in solids to their macroscopic properties including electrical, magnetic and thermal. You will be able to connect the microscopic and macroscopic pictures of the thermal properties of solids, and to describe the quantum statistics of degenerate Fermi gases, Bose-Einstein condensation, superfluidity in liquid helium and black body radiation.

Learn the fundamentals of quantum theory and how it applies to concrete physical systems. The module begins by establishing a basic working knowledge of nonrelativistic quantum mechanics based on the Schrödinger equation. You will develop skills necessary to apply quantum mechanics to simple, exactly solvable problems, as well as finding approximate descriptions of more complex quantum systems. This will enable you to make precise predictions for the behaviour of realistic quantum systems, and to understand the significance of the predictions for experimental observations. As a major application, you will use these skills to describe the basic characteristics of atomic structure, the process

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series. This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory. The cornerstone of mathematical analysis is the construction

Through a series of lectures, workshops, hands-on coursework and numerical simulations, gain command of advanced theoretical tools and mathematical methods for modelling and simulating many-body quantum systems. Specifically, you will learn the second quantisation and path integral methods of theoretical physics, supported by the mathematics of complex analysis. This enables you to apply basic concepts and techniques of field theory and its approximation methods. The workshops will provide a playground for hands-on use of these field theory techniques, culminating in an independent investigative simulation of a quantum system.

Explore the topics of relativity, nuclear physics and particle physics and learn about the principles of special relativity, Lorentz transformations and four-vectors. You will use relativity to understand topics such as particle decays and the doppler effect. In nuclear physics, you will study properties of nuclei, radioactive decay, nuclear fission and fusion. In particle physics, you will learn about the Standard Model, particle interactions, the structure of matter and modern particle physics experiments. Gain an understanding of the fundamental building blocks of the universe.

As part of a group, and supported by a supervisor, apply your mathematical or computer modelling techniques to an open-ended investigation of either real-world applications or models in theoretical physics or mathematics. Projects vary from year to year; examples of recent projects include: using cellular automata in cryptography, music composition or to model dynamical systems (traffic or fluid flow, fire or disease spreading, etc) machine learning applied to stock market data, predicting game outcomes or clustering of exoplanets data modelling quantum computers with applications to quantum game theory or quantum simulation simulation of chaotic dynamics in classical and quantum pendulums,

Commutative rings generalise both integers and polynomials and they play a very important role in a wide area of mathematics. As well as being important in algebra, they sit at the heart of algebraic approaches including geometry and number theory, in part because rings of functions occur so naturally there, as they do in analysis. At this stage, you will already know how to factor and divide integers and polynomials. Therefore, a crucial question is to understand the factorisability and divisibility properties in more general commutative rings. For example, what is the analogue of the set of prime integers, or which are the invertible elements? You will seek to answer these questions, begin

Explore the physics of semiconductors, superconductors and magnetic materials. You will learn how the properties of these materials arise from their microscopic structure and from interactions between atoms, electrons and phonons. You will gain a deeper understanding of the role of statistical concepts in understanding macroscopic systems and be able to solve selected model problems using advanced methods from condensed matter theory.

Cosmology treats the entire universe as a physical system. Investigate the evolution of the universe from the big bang, the cataclysmic explosion which started it all, through to its final fate in the distant future. Further areas of study include the dynamics of the universe as a whole, the initial singularity from which it originated, and the violent history of the early universe. You’ll also investigate the cosmic microwave background, the formation of structure (such as galaxies and galaxy clusters), cosmological horizons, dark matter and dark energy. You’ll consider the origin of matter, the age of the universe and the reason why is the sky is dark at night as you explore the limits of

Investigate the importance of conservation laws and discrete symmetries in particle interactions and experiments used to determine them. You will explore topics such as quark mixing, heavy-flavour physics, neutrino oscillations and particle interactions with matter. You will then use your knowledge of particle interactions with matter to understand the design of particle detectors and accelerators. Results and measurements from particle physics experiments will be used throughout the module to highlight the topics presented.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance. You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proo

An inner product space is a real or complex vector space, equipped with certain extra structure that formalises the geometrical notion of orthogonality. It turns out that each inner product space has an intrinsic notion of distance, allowing us to discuss convergence and completeness. Complete inner product spaces are known as Hilbert spaces. The theory of Hilbert spaces blends linear algebra and (real) analysis. It is a natural and powerful tool for studying problems of quantitative approximation. Furthermore, it provides an abstract framework that can be applied to diverse areas of maths, from differential equations and spectral theory to quantum mechanics and stochastic processes. This mo

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to quantum field theory and protein-folding. Develop tools to measure knottedness, including geometrical ideas like curvature, knot invariants like the Jones polynomial, and the crucial concept of the fundamental group, which has applications in topology far beyond detecting knots.

The module commences by looking at classical methods of encryption, discussing their advantages, disadvantages and efficiency. You will also investigate statistical attacks on these methods of encryption and the need for better methods. After this, you will explore modern methods of encryption that are used in the real-world and rely on the robustness of modular arithmetic. While most encryption methods are still considered secure, you will review potential attacks on these systems (e.g. factorisation algorithms) and situations where bad key generation or implementation has occurred. Production of a big enough quantum computer renders the above schemes useless. Therefore, you will dive into

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits. Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlo

This module is based on the most current research activity within the Department of Physics and is updated each year in line with current developments in the field. You will develop your expertise in applying the latest underlying physical ideas and concepts to formulate and tackle problems and improve your qualitative understanding of research frontiers. Possible topics include methods of advanced statistical mechanics, quantum field theory, geometric phases and topology, and differential geometry.

This individual project forms the exciting core of Year 4. Working across the year, you will experience cutting-edge research and complete a major open-ended investigation. The projects we offer are diverse and updated every year in line with current research activities, making use of available facilities and datasets. They are taught through individual supervision and training seminars to further develop your research and communication skills. To start the project, you will research the area and prepare a literature review. From this foundation you will plan, manage and execute your own investigative work. Throughout the project, you will use and develop your abilities to synthesise technic

Learn how complex phenomena in condensed matter physics are applied to create useful solid-state devices and nanoscale structures. You will explore physical concepts related to quantum transport and states of matter in low-dimensional systems and employ appropriate theoretical tools to describe their electronic behaviour. By the end of this module, you will be able to demonstrate an understanding of the physical effects underpinning the characteristics of common nanoscale devices, including quantum dots and nanomechanical structures, and discuss methods used in their fabrication and characterisation.

Combinatorics is a core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous, such as graphs, lattices, designs and codes. While combinatorics is a huge subject, with deep and important connections to many areas of modern mathematics, it is a very accessible one. Explore the fundamental topics of combinatorial enumeration (sophisticated counting methods) and combinatorial design theory (Latin squares and block designs). Alongside this, you will learn additional combinatorial topics chosen from areas such as set systems, error-detecting and error-correcting codes, and combinatorial geometry. Throughout the modu

Galois theory is the study of roots of polynomials and symmetries of these roots. A basic example of such a symmetry is complex conjugation, which swaps the roots of the polynomial x^2+1 (or any irreducible real quadratic). Polynomials of a degree higher than 2 can have much more complicated symmetries. In fact, any finite group is possible! Galois theory provides you with a framework for understanding how the group of symmetries encodes very deep information about the polynomial itself. A famous application, which will be covered in the module, is a proof of the Abel-Ruffini theorem. Unlike lower degrees, a general polynomial of degree 5 or higher has no ‘solution in radicals’ (i.e. obtaina

We introduce you to Einstein's theory of General Relativity, which is our current understanding of gravity, exploring the relationship between Newtonian mechanics and relativity. In general relativity there is no long-range gravitational force, only a local response to the curvature of spacetime. You will study black holes, gravitational waves, and cosmological phenomena such as cosmic inflation and the big bang. Key topics include the Einstein field equations, gravitational redshift and lensing, orbital precession, and relativistic electromagnetism. In this module you will also explore advanced concepts such as Hawking radiation, black hole thermodynamics and traversable wormholes. By the e

Dive into transformations and corresponding symmetries, group theory and its applications in particle physics, and the basics of the gauge-invariant field theories. You will use these topics to understand the properties of the quarks and leptons and their strong, electromagnetic and weak interactions. You will explore topics such as spontaneous symmetry breaking and the Higgs mechanism. This module will describe the foundations of the Standard Model of particle physics and its possible extensions.

The first time you meet groups, they tend to be finite: the symmetry group of a triangle, or a cube, or a permutation group. Lie theory is the study of continuous groups of transformations, like rotations of 2-, 3-, 4- or higher-dimensional spaces. It underpins most of modern geometry and particle physics, with applications from solving differential equations, to understanding matter made of quarks, to classifying polynomials in pure algebra. You will explore the foundations of this powerful subject and gain the skills to perform the complex calculations needed to understand the applications.

Have you ever wondered how to define the size of an unusually shaped subset of Euclidean space, such as the Cantor set? Or how to compute the integral of a function that is not piecewise continuous? This module will introduce the concept of a measure of a set, that generalises the idea of the length of an interval or the area of a rectangle to a much bigger class of sets - the measurable sets. You will then be able to generalise the idea of integrals of functions and lead to new methods for computing integrals. You will develop this theory using countability arguments, set theory, openness and closedness, as well as sequences of numbers or functions. You will learn to explore this theory usi

An introduction to analytic and algebraic techniques for studying problems in number theory. You will explore how methods for analysis can be useful in studying the distribution of prime numbers, the asymptotics of arithmetic functions and the solubility of Diophantine equations. You will also study the solubility of Diophantine equations using methods from algebra, particularly the concept of unique factorisation in certain rings (or lack of). By the end of the module, you will have developed a working knowledge of analytic and algebraic number theory.

Operator theory can be seen as an infinite-dimensional version of matrix theory, where matrices act linearly on vectors, they can be added and multiplied, and one can determine eigenvalues and Jordan normal forms etc. But, while some things are similar to finite dimensions, others are completely different, for example not every symmetric operator can be diagonalised. Due to infinite dimensionality, delicate convergence and completeness issues arise and require input from analysis. If you enjoyed both linear algebra and real analysis, this module is for you. It will prepare you for further postgraduate study in operator algebras, differential operators, mathematical quantum mechanics and othe

Develop an analytical and axiomatic approach to the theory of probabilities. First, you will examine the notion of a probability space through simple examples featuring both discrete and continuous sample spaces. Using random variables, you will develop a probability calculus which can be applied to achieve laws of large numbers for sums of independent random variables. You will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.