By Gerald W Johnson, Michel L. Lapidus, Lance Nielsen
This ebook is geared toward supplying a coherent, basically self-contained, rigorous and entire summary thought of Feynman's operational calculus for noncommuting operators. even though it is galvanized by way of Feynman's unique heuristic feedback and time-ordering principles in his seminal 1951 paper An operator calculus having functions in quantum electrodynamics, as may be made abundantly transparent within the creation (Chapter 1) and in other places within the textual content, the speculation constructed during this e-book additionally is going well past them in a few instructions which have been now not expected in Feynman's paintings. accordingly, the second one a part of the most name of this book.
The simple houses of the operational calculus are built and sure algebraic and analytic homes of the operational calculus are explored. additionally, the operational calculus could be visible to own a few friendly balance houses. moreover, an evolution equation and a generalized necessary equation obeyed through the operational calculus are mentioned and connections with sure analytic Feynman integrals are famous.
This quantity is basically self-contained and we simply imagine that the reader has an inexpensive, graduate point, history in research, degree thought and sensible research or operator conception. a lot of the mandatory final history is available in the textual content itself.
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Extra resources for Feynman’s Operational Calculus and Beyond: Noncommutativity and Time-Ordering
Example text
N eı ζ ,A fˆ(ζ ) dζ . The chapter ends with a discussion of various Trotter-like product formulas that can be obtained in the Banach space setting. Finally, in the epilogue to this book, Chapter 11, we present several open problems and suggestions for future research connected to various aspects of the theory of Feynman’s operational calculus developed in this monograph. The problems discussed involve possible extensions of the theory to noncommuting unbounded linear operators, general Borel measures (possibly with countably supported discrete parts) and a search for the corresponding evolution equations.
N, it is shown that H commutes with the disentangling map. 3 establishes first-order expansions by making use of the so-called derivation formula: If f ∈ D and if D is an arbitrary derivation of L(X), then D Tμ1 , . . , μn f A˜ 1 , . . , A˜ n n = Tμ1 , . . , μj ,μj ,μj+1 , . . , μn Fj A˜ 1 , . . , A˜ j , D(Aj ), A˜ j+1 , . . , A˜ n , j=1 where Fj (x1 , . . , xj , y, xj+1 , . . , xn ) := (∂/∂xj )f (x1 , . . , xn )y and μ1 , . . , μn are continuous probability measures. This formula is used to determine an expression for the operator C1 in the expansion f (A + B) = f (A) + C1 + 2 C2 + · · · + O( n ) as → 0+ .
N, k f A˜ 1, k , . . , A˜ n, k , with the arguments of f also depending on the sequential index k. So, for each value of k ∈ N, we are selecting an element of the operational calculus indexed by the n-tuple (μ1, k , . . , μn, k ). A brief outline of the four sections of Chapter 7 follows. 1 is concerned with the general setting for the stability theory. It is here that the reader will find the necessary definitions and theorems which form the underlying structure for the later sections. 2, we investigate joint stability in both the time-dependent and the time independent setting.