# How to be causal: time, spacetime and spectra

@article{Kinsler2011HowTB, title={How to be causal: time, spacetime and spectra}, author={Paul Kinsler}, journal={European Journal of Physics}, year={2011}, volume={32}, pages={1687 - 1700} }

I explain a simple definition of causality in widespread use, and indicate how it links to the Kramers–Kronig relations. The specification of causality in terms of temporal differential equations then shows us the way to write down dynamical models so that their causal nature in the sense used here should be obvious to all. To extend existing treatments of causality that work only in the frequency domain, I derive a reformulation of the long-standing Kramers–Kronig relations applicable not only…

## 25 Citations

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## References

SHOWING 1-10 OF 58 REFERENCES

### Causality and the Dispersion Relation: Logical Foundations

- Computer Science
- 1956

A rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part.

### Is There an Independent Principle of Causality in Physics?

- PhilosophyThe British Journal for the Philosophy of Science
- 2009

Mathias Frisch has argued that the requirement that electromagnetic dispersion processes are causal adds empirical content not found in electrodynamic theory. I urge that this attempt to reconstitute…

### What did Kramers and Kronig do and how did they do it?

- Physics
- 2010

Over time the account of how the Kramers–Kronig (dispersion) relations between the real and imaginary parts of response functions were derived in 1926 and 1927 has been transmogrified into anecdotes…

### Nature of time and causality in Physics

- Physics
- 2007

The conceptual definition and understanding of the nature of time, both qualitatively and quantitatively is of the utmost difficulty and importance, and plays a fundamental role in physics. Physical…

### Electromagnetism and time-asymmetry

- Philosophy, Geology
- 2010

It is a commonplace to note that in a world governed by special or general relativity, an observer has access only to data within her past lightcone (if that). The significance of this for…

### Causality-based criteria for a negative refractive index must be used with care.

- PhysicsPhysical review letters
- 2008

Using the principle of causality as expressed in the Kramers-Kronig relations, we derive a generalized criterion for a negative refractive index that admits imperfect transparency at an observation…

### Causality: Models, Reasoning and Inference

- Philosophy
- 2000

1. Introduction to probabilities, graphs, and causal models 2. A theory of inferred causation 3. Causal diagrams and the identification of causal effects 4. Actions, plans, and direct effects 5.…

### Presenting electromagnetic theory in accordance with the principle of causality

- Geology
- 2004

A method of presenting electromagnetic theory in accordance with the principle of causality is described. Two 'causal' equations expressing time-dependent electric and magnetic fields in terms of…

### Space, Time, and Causality: An Essay in Natural Philosophy

- Philosophy
- 1985

If looking for the book by J. R. Lucas Space, Time, and Causality: An Essay in Natural Philosophy in pdf format, then you've come to the faithful website. We furnish full option of this ebook in doc,…

### A formal interpretation of the displacement current and the instantaneous formulation of Maxwell’s equations

- Physics
- 2011

Maxwell’s displacement current has been the subject of controversy for more than a century. Questions on whether the displacement current represents a true current like the conduction current and…