Additions:
Additions:
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the [[http://en.wikipedia.org/wiki/Inverse_Fourier_transform Fourier inversion theorem]]. In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over ""p"" and over ""x"" (or ""y"") really stand for finite sums with ""N"" terms, where ""N"" is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which ""N"" goes to infinity, provided that ""N"" is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in [[QuantumCovariance Quantum Covariance]] and in [[Regularisation Regularisation]].
Deletions:
Thus, we can precisely recover the coefficients of position if we know the momentum space wave function at any time. This is the [[http://en.wikipedia.org/wiki/Inverse_Fourier_transform Fourier inversion theorem]]. In standard quantum theory there are some subtle mathematical problems with Fourier inversion, because of the use of continuous transforms. If we remember that the integrals over ""p"" and over ""x"" (or ""y"") really stand for finite sums with ""N"" terms, where ""N"" is large and the same in both sums, these problems never arise. There are mathematical issues to resolve concerning discreteness and Lorentz transformation. These issues do not arise for discrete transforms, and do not appear in the limit in which ""N"" goes to infinity, provided that ""N"" is kept the same in both sums. One may then recover a continuum theory without divergence problems. This willl be considered in more depth in [[QuantumCovariance Quantum Covariance]] and in [[DiscreteQuantumElectrodynamics Discrete Quantum Electrodynamics]].
Additions:
====""""Quantum Logic====
====""""Truth Values====
====""""Sentences Describing Hypothetical Measurement Results====
====""""Integral Representation====
====""""Plane Wave States====
====""""The Resolution of Unity in Momentum Space====
Deletions:
====""""Quantum Logic====
====""""Truth Values====
====""""Sentences Describing Hypothetical Measurement Results====
====""""Integral Representation====
====""""Plane Wave States====
====""""The Resolution of Unity in Momentum Space====
Additions:
====""""Quantum Logic====
====""""Truth Values====
""R(x)"" is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. Its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like ""R(x)"", in the subjunctive mood.
====""""Sentences Describing Hypothetical Measurement Results====
The conditional clause, referring to the first measurement, is represented by a ket. It is described as a //formal conditional clause// to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The //basic conditional clauses//, on which the language is built, refer to measurements of position:
The set of formal conditional clauses, or kets, now has the mathematical structure of an ""N""-dimensional "" vector space"". The symbol ""H1(t)"" is used to denote this vector space, and consists of a family of formal conditional clauses concerning the measurement of position of a single particle at time ""t"". The basic conditional clauses, ""
"", described in rule I are a basis for ""H1(t)"".
====""""Integral Representation====
====""""Plane Wave States====
<<""Definition: For each ket, 
, define the momentum space wave function, F (p)"", such that
====""""The Resolution of Unity in Momentum Space====
Deletions:
====""""Quantum Logic====
====""""Truth Values====
""R(x)"" is intuitively sensible, even when no measurement is done, but it is neither strictly true nor false, and cannot sensibly be given a crisp truth value. its truth is distinguished from a probability because, when no measurements are to be done, we cannot sensibly discuss the potential frequency of individual measurement results. In quantum theory we are not always going to do a measurement, but we still want to talk about what would happen if we were to do a measurement, i.e. we need to be able to make statements about hypothetical measurement results. Quantum logic provides a way of discussing levels of truth for statements about hypothetical measurement, like ""R(x)"", in the subjunctive mood.
====""""Sentences Describing Hypothetical Measurement Results====
The conditional clause, referring to the first measurement, is represented by a ket. It is described as a //formal conditional clause// to indicate that only clauses formally described in the rules are allowed as part of the logic, or formal language. The basic conditional clauses, on which the language is built, refer to measurements of position:
The set of formal conditional clauses, or kets, now has the mathematical structure of an ""N""-dimensional "" vector space"". The symbol ""H1(t)"" is used to denote this vector space, and consists of a family of statements we can make about the measurement of position of a single particle at time ""t"". The basic conditional clauses, """", described in rule I are a basis for ""H1(t)"".
====""""Integral Representation====
====""""Plane Wave States====
<<""Definition: For each ket, , define the momentum space wave function, f (p)"", such that
====""""The Resolution of Unity in Momentum Space====
