The mathematical physical equations satisfied by retarded and advanced Green’s functions
Bibliographic record
Abstract
In mathematical physics, time-dependent Green’s functions (GFs) are the solutions of differential equations of the first and second time derivatives. Habitually, the time-dependent GFs are Fourier transformed into the frequency space. Then, analytical continuation of the frequency is extended to below or above the real axis. After inverse Fourier transformation, retarded and advanced GFs can be obtained, and there may be arbitrariness in such analytical continuation. In the present work, we establish the differential equations from which the retarded and advanced GFs are rigorously solved. The key point is that the derivative of the time step function is the Dirac δ function plus an infinitely small quantity, where the latter is not negligible because it embodies the meaning of time delay or time advance. The retarded and advanced GFs defined in this paper are the same as the one-body GFs defined with the help of the creation and destruction operators in many-body theory. There is no way to define the causal GF in mathematical physics, and the reason is given. This work puts the initial conditions into differential equations, thereby paving a way for solving the problem of why there are motions that are irreversible in time.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".