A mathematical interpretation of Dirac's formalism. Part C: Free field operators

S.J.L. Eijndhoven, van; J. Graaf, de

doi:10.1016/0034-4877(86)90026-1

A mathematical interpretation of Dirac's formalism. Part C: Free field operators

S.J.L. Eijndhoven, van, J. Graaf, de

Center for Analysis, Scientific Computing and Applications

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Abstract

In the present paper we give a mathematical regorization and interpretation of the canonical (anti-) commutation relations which occur in the theory of the quantized free field. In our set-up, with respect to the physics, we felt inspired by the paper [Ro] of B. Robertson. His paper covers all essential elements of the subject under consideration. The main part of [Ro] will be mathematically justified and even generalized. In this connection we mention also the monograph [Sh] of A.S. Shvarz. With respect to the involved mathematics the papers [Co] of J. M. Cook and [KMP] of P. Kristensen et al. have been a source of inspiration. Our set-up is closely related to two of our earlier papers [EG 1–2] in which we presented a mathematical interpretation of Dírac's formalism. This interpretation is based on two new mathematical concepts: the concept of Dirac basis ans the concept of bracket. The bracket is no longer regarded as a "number" but as an analytic function on the open right half of the complex plane. As in [EG 1–2], we use the theory of generalized functions which has been constructed by one of us (cf. [G]). Quantum theory and distribution theory seem to be dissolubly connected. One might wonder whether the choice of the distribution theory plays an essential role in a mathematical interpretation of Dirac's formalism and related physical concepts. The answer is threefold: There are strong indications that similar interpretations can be derived in a very wide class of distribution theories (which include Schwarz's theory of tempered distributions). This class has been described in our papers [EK] and [EGK]. On the other hand, the distribution theory employed here, introduces generalized functions in a way very close to the physical intuitive view on improper functions. Moreover, it seems natural to adapt the used distribution theory to the concrete quantum mechanical system under consideration. Mathematical subject codes: 46 F 10; 47 A 70; 81 B 05

Original language	English
Pages (from-to)	299-326
Number of pages	28
Journal	Reports on Mathematical Physics
Volume	23
Issue number	3
DOIs	https://doi.org/10.1016/0034-4877(86)90026-1
Publication status	Published - 1986

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title = "A mathematical interpretation of Dirac's formalism. Part C: Free field operators",

abstract = "In the present paper we give a mathematical regorization and interpretation of the canonical (anti-) commutation relations which occur in the theory of the quantized free field. In our set-up, with respect to the physics, we felt inspired by the paper [Ro] of B. Robertson. His paper covers all essential elements of the subject under consideration. The main part of [Ro] will be mathematically justified and even generalized. In this connection we mention also the monograph [Sh] of A.S. Shvarz. With respect to the involved mathematics the papers [Co] of J. M. Cook and [KMP] of P. Kristensen et al. have been a source of inspiration. Our set-up is closely related to two of our earlier papers [EG 1–2] in which we presented a mathematical interpretation of D{\'i}rac's formalism. This interpretation is based on two new mathematical concepts: the concept of Dirac basis ans the concept of bracket. The bracket is no longer regarded as a {"}number{"} but as an analytic function on the open right half of the complex plane. As in [EG 1–2], we use the theory of generalized functions which has been constructed by one of us (cf. [G]). Quantum theory and distribution theory seem to be dissolubly connected. One might wonder whether the choice of the distribution theory plays an essential role in a mathematical interpretation of Dirac's formalism and related physical concepts. The answer is threefold: There are strong indications that similar interpretations can be derived in a very wide class of distribution theories (which include Schwarz's theory of tempered distributions). This class has been described in our papers [EK] and [EGK]. On the other hand, the distribution theory employed here, introduces generalized functions in a way very close to the physical intuitive view on improper functions. Moreover, it seems natural to adapt the used distribution theory to the concrete quantum mechanical system under consideration. Mathematical subject codes: 46 F 10; 47 A 70; 81 B 05",

author = "{Eijndhoven, van}, S.J.L. and {Graaf, de}, J.",

year = "1986",

doi = "10.1016/0034-4877(86)90026-1",

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T1 - A mathematical interpretation of Dirac's formalism. Part C: Free field operators

AU - Eijndhoven, van, S.J.L.

AU - Graaf, de, J.

PY - 1986

Y1 - 1986

N2 - In the present paper we give a mathematical regorization and interpretation of the canonical (anti-) commutation relations which occur in the theory of the quantized free field. In our set-up, with respect to the physics, we felt inspired by the paper [Ro] of B. Robertson. His paper covers all essential elements of the subject under consideration. The main part of [Ro] will be mathematically justified and even generalized. In this connection we mention also the monograph [Sh] of A.S. Shvarz. With respect to the involved mathematics the papers [Co] of J. M. Cook and [KMP] of P. Kristensen et al. have been a source of inspiration. Our set-up is closely related to two of our earlier papers [EG 1–2] in which we presented a mathematical interpretation of Dírac's formalism. This interpretation is based on two new mathematical concepts: the concept of Dirac basis ans the concept of bracket. The bracket is no longer regarded as a "number" but as an analytic function on the open right half of the complex plane. As in [EG 1–2], we use the theory of generalized functions which has been constructed by one of us (cf. [G]). Quantum theory and distribution theory seem to be dissolubly connected. One might wonder whether the choice of the distribution theory plays an essential role in a mathematical interpretation of Dirac's formalism and related physical concepts. The answer is threefold: There are strong indications that similar interpretations can be derived in a very wide class of distribution theories (which include Schwarz's theory of tempered distributions). This class has been described in our papers [EK] and [EGK]. On the other hand, the distribution theory employed here, introduces generalized functions in a way very close to the physical intuitive view on improper functions. Moreover, it seems natural to adapt the used distribution theory to the concrete quantum mechanical system under consideration. Mathematical subject codes: 46 F 10; 47 A 70; 81 B 05

AB - In the present paper we give a mathematical regorization and interpretation of the canonical (anti-) commutation relations which occur in the theory of the quantized free field. In our set-up, with respect to the physics, we felt inspired by the paper [Ro] of B. Robertson. His paper covers all essential elements of the subject under consideration. The main part of [Ro] will be mathematically justified and even generalized. In this connection we mention also the monograph [Sh] of A.S. Shvarz. With respect to the involved mathematics the papers [Co] of J. M. Cook and [KMP] of P. Kristensen et al. have been a source of inspiration. Our set-up is closely related to two of our earlier papers [EG 1–2] in which we presented a mathematical interpretation of Dírac's formalism. This interpretation is based on two new mathematical concepts: the concept of Dirac basis ans the concept of bracket. The bracket is no longer regarded as a "number" but as an analytic function on the open right half of the complex plane. As in [EG 1–2], we use the theory of generalized functions which has been constructed by one of us (cf. [G]). Quantum theory and distribution theory seem to be dissolubly connected. One might wonder whether the choice of the distribution theory plays an essential role in a mathematical interpretation of Dirac's formalism and related physical concepts. The answer is threefold: There are strong indications that similar interpretations can be derived in a very wide class of distribution theories (which include Schwarz's theory of tempered distributions). This class has been described in our papers [EK] and [EGK]. On the other hand, the distribution theory employed here, introduces generalized functions in a way very close to the physical intuitive view on improper functions. Moreover, it seems natural to adapt the used distribution theory to the concrete quantum mechanical system under consideration. Mathematical subject codes: 46 F 10; 47 A 70; 81 B 05

U2 - 10.1016/0034-4877(86)90026-1

DO - 10.1016/0034-4877(86)90026-1

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SN - 0034-4877

VL - 23

SP - 299

EP - 326

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

IS - 3

ER -

A mathematical interpretation of Dirac's formalism. Part C: Free field operators

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