BosonQuest

Learn Physics or Get Bored Trying

Deviating standards: epilogue

A subtle variant of the “inconsistencies” in the last post can arise where a concept is introduced and used in a simplified form, which can catch you unawares in more general settings.  For example (and while I’m trying to keep these posts agnostic of any particular field, it’s hard to describe this without example; but don’t worry if the details aren’t clear, it’s the idea that matters):

A vector space has associated with it a scalar field.  The most obvious one is the field of real numbers; they are also commonly associated with complex numbers.  Given a vector space over either of these fields, you can turn this into an inner product space by defining the inner product of two vectors u and v as x = \langle u, v \rangle, where x is a member of the scalar field.  The inner product must satisfy a number of properties; one of these dictates a form of commutation: \langle u, v \rangle = \overline{\langle v, u \rangle}, where \overline{x} denotes complex conjugation—a process that leaves real numbers unchanged.  You will therefore, very occasionally, see an author avoid mentioning complex numbers if they are making only brief use of an inner product space with real scalars, by quoting this property as \langle u, v \rangle = \langle v, u \rangle.  If not aware of this, you may miss the fact that conjugation is required in the complex environment.

Again, the only real protection to this is to make sure you read any introductory chapters where terms ought to be properly defined.  If in doubt, and the text makes assumptions of prior knowledge, Wikipedia is generally a good place to look for definitions, as it will tend to quote the most general form.

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June 16, 2010 Posted by | Self-study | Leave a comment