QFT Scattering: Perturbation Vs. LSZ

by RICHARD 37 views

Hey everyone! Let's dive into a fascinating corner of Quantum Field Theory (QFT): scattering. Specifically, we're going to explore two primary methods for calculating the S-matrix, which, if you recall, describes the probabilities of various outcomes in scattering experiments. These methods are: perturbative calculations in the interaction picture and the LSZ reduction formula. Buckle up, because it's going to be a fun ride!

Perturbation Theory in the Interaction Picture

Let's kick things off with perturbation theory in the interaction picture. This approach is probably the one you're most familiar with, especially if you've worked with Feynman diagrams. The basic idea is to treat the interaction part of the Hamiltonian as a 'small' perturbation to the free theory. This allows us to expand the time evolution operator, and hence the S-matrix, as a power series in the interaction strength.

In this framework, the S-matrix can be expressed as:

S = T exp[-i ∫ dt H_I(t)]

where T is the time-ordering operator and H_I(t) is the interaction Hamiltonian in the interaction picture. Expanding the exponential, we get an infinite series:

S = 1 - i ∫ dt H_I(t) + (-i)²/2! ∫ dt ∫ dt' T{H_I(t)H_I(t')} + ...

Each term in this series corresponds to a different order of interaction. For example, the first term (1) represents no interaction at all, the second term represents a single interaction, the third term represents two interactions, and so on. These terms can be visually represented using Feynman diagrams, which provide a powerful tool for calculating scattering amplitudes.

Each Feynman diagram corresponds to a specific term in the perturbation series. The lines in the diagram represent propagators (virtual particles), and the vertices represent interactions. By applying Feynman rules, we can translate each diagram into a mathematical expression that contributes to the scattering amplitude. The total scattering amplitude is then obtained by summing the contributions from all possible diagrams at a given order in the perturbation series.

However, calculating the S-matrix using perturbation theory involves several important steps and considerations. First, we need to choose an appropriate interaction Hamiltonian. This choice depends on the specific theory we are considering. For example, in Quantum Electrodynamics (QED), the interaction Hamiltonian describes the interaction between electrons and photons. Second, we need to calculate the Feynman diagrams corresponding to the desired order in the perturbation series. This can be a challenging task, especially for higher-order diagrams, which can be numerous and complex. Third, we need to apply the Feynman rules to translate each diagram into a mathematical expression. This requires careful attention to detail and a good understanding of the underlying theory. Finally, we need to sum the contributions from all possible diagrams to obtain the total scattering amplitude. This can be a computationally intensive task, especially for theories with many particles and interactions.

Despite these challenges, perturbation theory is a powerful and widely used tool for calculating scattering amplitudes in QFT. It allows us to make predictions about the behavior of particles at high energies and to test the validity of our theoretical models against experimental data. Moreover, the use of Feynman diagrams provides a visually intuitive way to understand the underlying physics of scattering processes.

LSZ Reduction Formula

Now, let's switch gears and talk about the LSZ reduction formula. This is a more formal approach to calculating the S-matrix, rooted in the fundamental principles of QFT. It relates the S-matrix elements directly to the time-ordered correlation functions of the quantum fields.

The LSZ reduction formula provides a rigorous connection between the S-matrix elements and the vacuum expectation values of time-ordered products of field operators. The general form of the LSZ reduction formula for a 2-to-2 scattering process is:

<p_1, p_2 out | k_1, k_2 in> = ∫ dx_1 dx_2 dy_1 dy_2 e^(i(k_1.x_1 + k_2.x_2 - p_1.y_1 - p_2.y_2)) (□{x_1} + m²) (□{x_2} + m²) (□{y_1} + m²) (□{y_2} + m²) <0|T{φ(x_1)φ(x_2)φ(y_1)φ(y_2)}|0>

Where:

  • p_i are outgoing momenta
  • k_i are incoming momenta
  • □ is the d'Alembertian operator
  • m is the mass of the field
  • φ(x) is the field operator
  • T denotes time ordering

The beauty of the LSZ formula is that it expresses the S-matrix element in terms of the vacuum expectation value of a time-ordered product of field operators. This vacuum expectation value can be calculated using various techniques, such as perturbation theory or non-perturbative methods. The LSZ formula provides a rigorous connection between the S-matrix and the underlying quantum field theory, and it is particularly useful for studying scattering processes in theories with strong interactions or bound states.

The LSZ reduction formula is derived by carefully considering the asymptotic behavior of the quantum fields. In the far past and far future, the interacting fields behave like free fields. This allows us to express the in- and out-states in terms of these asymptotic fields. By inserting these expressions into the definition of the S-matrix, we arrive at the LSZ reduction formula. The formula involves taking derivatives with respect to time and spatial coordinates, which effectively 'peel off' the external legs of the Feynman diagrams.

One of the key advantages of the LSZ reduction formula is that it is valid even when perturbation theory breaks down. This is because it is based on the fundamental principles of QFT, such as unitarity and causality, rather than on a perturbative expansion. As a result, the LSZ reduction formula can be used to study scattering processes in theories with strong interactions or bound states, where perturbation theory is not applicable. However, the LSZ reduction formula can also be more challenging to apply in practice, especially for theories with many particles and interactions. The calculation of the vacuum expectation values of time-ordered products of field operators can be a complex task, and it often requires the use of advanced techniques from QFT.

Perturbation Theory vs. LSZ: A Comparison

So, which method is better? Well, it depends! Perturbation theory is generally easier to apply in practice, especially for simple theories with weak interactions. It provides a clear and intuitive way to calculate scattering amplitudes using Feynman diagrams. However, it is only valid when the interaction strength is small. When the interaction strength is large, perturbation theory breaks down, and we need to resort to other methods.

The LSZ reduction formula, on the other hand, is more formal and rigorous. It is valid even when perturbation theory breaks down. However, it can be more challenging to apply in practice, especially for complex theories. The LSZ reduction formula is often used as a starting point for developing non-perturbative methods for calculating scattering amplitudes.

Here's a quick rundown:

  • Perturbation Theory:
    • Pros: Intuitive, easier to calculate (usually).
    • Cons: Only valid for weak interactions.
  • LSZ Reduction:
    • Pros: Rigorous, valid even for strong interactions.
    • Cons: More complex, can be difficult to apply.

In many practical situations, the two methods are complementary. Perturbation theory can be used to obtain approximate results, while the LSZ reduction formula can be used to check the validity of these results and to develop more accurate methods.

Think of it like this: if you're trying to build a simple model airplane (weak interaction), perturbation theory is your easy-to-use instruction manual. But if you're designing a real aircraft (strong interaction), you need the rigorous engineering principles embodied by the LSZ reduction formula.

Conclusion

Both perturbative calculations and the LSZ reduction formula are essential tools in the QFT toolkit for understanding scattering processes. While perturbation theory offers a more accessible and intuitive approach for weakly interacting theories, the LSZ reduction formula provides a rigorous foundation and remains valid even when interactions become strong. Choosing the right method, or combining both, depends on the specific problem you're tackling. Keep exploring, keep questioning, and keep pushing the boundaries of our understanding of the quantum world! Happy scattering, folks! This exploration hopefully gives you a solid foundation in QFT Scattering!