Comments on Entire Functions of the Derivative Operator
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912911340863488 |
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| author | Woodard, R. P. |
| author_facet | Woodard, R. P. |
| contents | Many attempts to introduce fundamental nonlocality into quantum (or classical) field theory are based on the assumption that exponentials of the d'Alembertian are positive-definite, so that these operators can be employed without engendering the Ostrogradskian instability associated with higher derivative Lagrangians. {\bf This assumption is false.} Working in the simple context of a 1-dimensional, point particle $q(t)$, I demonstrate that the equation $\exp[T^2 \tfrac{d^2}{dt^2}] q(t) = 0$ has an infinite number of rapidly oscillating, exponentially rising and falling solutions. This infinite kernel is in one-to-one correspondence with the ability to specify ``initial value data'' {\it arbitrarily} over {\it any} finite interval $t_1 < t < t_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Comments on Entire Functions of the Derivative Operator Woodard, R. P. General Relativity and Quantum Cosmology High Energy Physics - Theory Many attempts to introduce fundamental nonlocality into quantum (or classical) field theory are based on the assumption that exponentials of the d'Alembertian are positive-definite, so that these operators can be employed without engendering the Ostrogradskian instability associated with higher derivative Lagrangians. {\bf This assumption is false.} Working in the simple context of a 1-dimensional, point particle $q(t)$, I demonstrate that the equation $\exp[T^2 \tfrac{d^2}{dt^2}] q(t) = 0$ has an infinite number of rapidly oscillating, exponentially rising and falling solutions. This infinite kernel is in one-to-one correspondence with the ability to specify ``initial value data'' {\it arbitrarily} over {\it any} finite interval $t_1 < t < t_2$. |
| title | Comments on Entire Functions of the Derivative Operator |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2602.16190 |