Both the mathematics of quantum field theory (QFT) and the physics of condensed matter are highly challenging and require a strong mathematical background. However, comparing the two in terms of toughness is subjective and depends on an individual's strengths, interests, and background.
Quantum field theory is a theoretical framework that combines quantum mechanics with special relativity to describe the behavior of elementary particles and their interactions. It involves advanced mathematical concepts such as functional analysis, operator algebras, group theory, and differential geometry. QFT can be quite abstract and requires a deep understanding of mathematical structures and techniques.
On the other hand, condensed matter physics deals with the properties and behavior of matter in its condensed phase, such as solids and liquids. It involves studying emergent phenomena, quantum effects, and the interactions of many particles. The physics of condensed matter can be highly complex due to the rich variety of systems and phenomena involved, including superconductivity, magnetism, topological phases, and many-body physics. The mathematical tools used in condensed matter physics include quantum mechanics, statistical mechanics, solid-state physics, and various mathematical techniques like group theory, tensor calculus, and numerical methods.
The perceived toughness of these fields can vary depending on the individual's mathematical background, previous exposure, and personal interests. Some individuals may find the abstract mathematical formalism of QFT more challenging, while others may struggle with the complexity and diverse phenomena of condensed matter physics. It is also worth noting that both fields are actively researched and continue to evolve, presenting new challenges and areas of exploration.
Ultimately, determining which field is tougher is subjective and can vary from person to person. Both QFT and condensed matter physics require significant dedication, a solid understanding of mathematics and physics, and a willingness to engage in rigorous problem-solving and abstract thinking.