For those who enjoy exploring, one of the things you eventually encounter is whole universe of ★ operators (e.g. star operators) that is sometimes not fully explained and frequently confusing as not all ★ operators. Adding to the confusion is the lack of notational formalism amongst authors. In many cases you will see the use of a simple asterisk * for the star operation, not to be confused with multiplication, as well as the larger asterisk and then also the star itself.
There are three specific instances of the use of the ★ operators that one should keep separate in their minds which are frequently context dependent. These are:
Moyal product – which defines an associative, non-commutative, algebra for functions and plays a central role in Wigner-Weyl transformations and our use of the Wigner quasi-probability distribution in phase space transformations into Hilbert spaces
Hodge Star – which defines an operation where if one has a k-dimensional vector representation of a larger n-dimensional space, one can find a dual representation in the n-k dimensional space – e.g. it is a transformation operation of the basis for some function
I don’t have time this morning to go into too much detail, so I’ll have to promise a follow up post, but one concept that we have touched on before in our mention of wedge products is the concept of differential forms:
For instance, the expression f(x) dx from one-variable calculus is called a 1-form, and can be integrated over an interval [a,b] in the domain of f:
and similarly the expression:
is a 2-form that has a surface integral over an oriented surface S:
Likewise, a 3-form represents a volume element that can be integrated over a region of space.
The definition of the differential form, combined with the use of the Hodge Star, allows us to develop a mathematical description of gauge theories in a more compact fashion, such as what is used to describe electromagnestism, where one can write the Maxwell equations compactly as:
Where F is the electromagnetic field strength, A is the vector potential, are viewed as structure constants and J is the current three-form that describes the current density in the various directions.
Electromagnetism is Abelian which means for the non-commutative Yang-Mills equation for field strength,
the wedge vanishes.
This is all important discussion, since it ties directly to our concepts on covariance as it relates to locality requirements in gauge theories, something we discussed in one of our earlier discussion about the Higgs; and is critical to our understanding of Matrix theory and approaches to finding non-perturbative, covariant forms of the theory (which at present do not exist). The Moyal product is important in understanding what is meant by “twisted convolution” which is basically defined by taking the inverse Fourier transform of the Moyal product of functions. One can use this to conceptualize a process of taking an ordinary phase space distribution, transforming it into Hilbert space and then determining the conjugate spectrum. These sorts of processes are conducted in hopes of finding a consistent means of localizing string field theory in terms of strings that are manifestly non-local and non-commutative, which is represented by taking the string coupling to value greater than zero, , when the theory is non-interactive. The ties between string interaction strength and string length are not trivial. Strings in some sense must have length in order to interact and vice versa.
What sets string theory apart from other theories is that the string interaction strength is not a constant but is a field in itself, thereby making string length a field as a well. This tells us that the a covariant form of Matrix theory must be very similar to gravity in that the metric tensor is also a field of metrics. What we do require is the gravitational constant to be constant, which means the string interaction strength and string length should be treated as conjugate variables. The gravitational constant is linked very closely to the Planck Length; the string interaction strength gets weaker as the string length grows larger than the Planck Length, this tells us we have to get our scale very close to the Plank Length before string interactions become relevant to physics.
That’s enough for today. As promised I will write some more at a later date.