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Quantum electrodynamic uncertainty relations
Volume 90A, number 6
PHYSICS LETTERS
19 July 1982
QUANTUM ELECTRODYNAMIC UNCERTAINTY RELATIONS A. WIDOM Physics Department, Northeastern University, Boston, MA 02115, USA
and TD. CLARK Physics Laboratory, University of Sussex, Brighton, Sussex, UK Received 17 March 1982 Revised manuscript received 13 May 1982
The canonical commutation relations between the electromagnetic fields E and B are stated in a form which is of use in the electrical engineering of magnetometers and electrometers. The result does not depend on gauge.
In our previous work on quantum electrodynamic engineering [1], the fact that E and B in overlapping regions of space exhibit uncertainty principle measurement interference is essential. The engineering notion of field flux actually simplifies the canonical commutation relations of quantum electrodynamics [2] in that the result does not depend on gauge. For a given surface S the electric and magnetic flux are defined respectively as 4~Q(S)= ffE. dt,
(1)
S
(F(S)
=
ff8. d~.
(2)
Now consider two surfaces S1 and S2 such that the boundary curve of one of the surfaces punctures the other surface a finite number of times. If one orients the surfaces and boundary curves such that the punctures are given a weight ±1 according to the direction, then a topological integer N(S1, S2) exists which is the algebraic sum of the weights.
280
The canonical quantisation of the electrodynamic field then reads [Q(S2), ‘F(S1)]
ihcN(S2, S1)
(3)
.
The resulting uncertainty principle between E and B measurements is evidently ~Q(S2)L~cF(S1) (hcI2) IN(S2, S1) I (4) The notion of using fields integrated over surfaces yields a more precise picture of electromagnetic field uncertainties than that of using volume averages. Finally, an engineering electrometer or magnetom~‘
.
eter [3] flux viewpoint is inherently gauge symmetric. No direct reference to gauge vector potentials is required, although vector potentials may be of use. References tii A. Widom,J. Low Temp. Phys. 37 (1979) 449. [2} Schiff, Quantum mechanics (McGraw-Hill, New York, 1955) Ch. XIV. 13] R.J. France et al., Nature 289 (1981) 543.
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
PHYSICS LETTERS
19 July 1982
QUANTUM ELECTRODYNAMIC UNCERTAINTY RELATIONS A. WIDOM Physics Department, Northeastern University, Boston, MA 02115, USA
and TD. CLARK Physics Laboratory, University of Sussex, Brighton, Sussex, UK Received 17 March 1982 Revised manuscript received 13 May 1982
The canonical commutation relations between the electromagnetic fields E and B are stated in a form which is of use in the electrical engineering of magnetometers and electrometers. The result does not depend on gauge.
In our previous work on quantum electrodynamic engineering [1], the fact that E and B in overlapping regions of space exhibit uncertainty principle measurement interference is essential. The engineering notion of field flux actually simplifies the canonical commutation relations of quantum electrodynamics [2] in that the result does not depend on gauge. For a given surface S the electric and magnetic flux are defined respectively as 4~Q(S)= ffE. dt,
(1)
S
(F(S)
=
ff8. d~.
(2)
Now consider two surfaces S1 and S2 such that the boundary curve of one of the surfaces punctures the other surface a finite number of times. If one orients the surfaces and boundary curves such that the punctures are given a weight ±1 according to the direction, then a topological integer N(S1, S2) exists which is the algebraic sum of the weights.
280
The canonical quantisation of the electrodynamic field then reads [Q(S2), ‘F(S1)]
ihcN(S2, S1)
(3)
.
The resulting uncertainty principle between E and B measurements is evidently ~Q(S2)L~cF(S1) (hcI2) IN(S2, S1) I (4) The notion of using fields integrated over surfaces yields a more precise picture of electromagnetic field uncertainties than that of using volume averages. Finally, an engineering electrometer or magnetom~‘
.
eter [3] flux viewpoint is inherently gauge symmetric. No direct reference to gauge vector potentials is required, although vector potentials may be of use. References tii A. Widom,J. Low Temp. Phys. 37 (1979) 449. [2} Schiff, Quantum mechanics (McGraw-Hill, New York, 1955) Ch. XIV. 13] R.J. France et al., Nature 289 (1981) 543.
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland