Both string theory and quantum field theory are highly complex and challenging areas of theoretical physics. It is difficult to definitively say which one is harder, as the difficulty can vary depending on the specific aspects being considered. Here are some points to consider:
Mathematical Complexity: String theory involves a high level of mathematical complexity. It requires advanced mathematical tools, such as differential geometry, topology, and algebraic geometry, to describe the behavior of strings and their interactions. Quantum field theory also involves intricate mathematical formalism, including the manipulation of fields, operators, and Feynman diagrams. Both theories require a strong mathematical background, making them challenging in their own right.
Conceptual Framework: String theory introduces new concepts and ideas compared to classical field theories. It postulates that fundamental particles are not point-like but are instead one-dimensional objects known as strings. This requires a shift in thinking and an understanding of concepts such as extra dimensions and string vibrations. Quantum field theory, on the other hand, builds upon the framework of quantum mechanics and extends it to fields. It requires a deep understanding of quantum phenomena and the interaction of particles and fields.
Experimental Validation: Currently, neither string theory nor quantum field theory has been experimentally confirmed in their complete forms. Quantum field theory, however, has been successfully applied to describe and predict particle interactions in the framework of the Standard Model of particle physics, which has been extensively validated experimentally. String theory, while it offers a compelling framework for unifying quantum mechanics and gravity, lacks direct experimental evidence and faces challenges in making testable predictions.
Research Focus: The research focus within each theory can also influence perceived difficulty. String theory encompasses a broad range of topics, including understanding the behavior of strings in different spacetime dimensions, exploring duality symmetries, and investigating the landscape of string vacua. Quantum field theory is a vast field with diverse research directions, such as quantum chromodynamics (QCD), electroweak theory, and quantum field theories in condensed matter physics.
In summary, both string theory and quantum field theory pose significant intellectual challenges and require deep mathematical and conceptual understanding. The difficulty may vary depending on the specific aspects under consideration and the researcher's background and expertise. It is worth noting that these theories are highly interconnected, with string theory incorporating quantum field theory as a framework in certain limits, and they are often studied in conjunction with each other.