Abstract
Problems with an exact construction of the quantum \(\phi ^4_4\) model. The interaction picture. The Gell-Mann–Low formula for Greeen’s functions. The generating functional for Green’s functions. The exponential Wick formula. The Feynman free propagator. Regularized Feynman diagrams in four-momentum space. Normal ordered interactions. Cancelation of vacuum bubbles.
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Notes
- 1.
We assume here that the considered operators do not depend on time in the Schroedinger picture.
- 2.
In the case of fields defined in two- or three-dimensional space-time the situation is a little bit better.
- 3.
Here we are simplifying things a little bit. In order to be sure that the state \(|\psi \rangle \) belongs to the Hilbert space one should integrate \( T\left( \hat{\phi }(x_1) \ldots \hat{\phi }(x_n) \right) \) with a test function \(h(x_1, x_2, \ldots , x_n)\). We are assuming that such a ‘technical’ step is done implicitly.
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Arodź, H., Hadasz, L. (2017). Perturbative Expansion in the \( \phi ^4_4\) Model. In: Lectures on Classical and Quantum Theory of Fields. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55619-2_7
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DOI: https://doi.org/10.1007/978-3-319-55619-2_7
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