The following limit result is well known: \(e^x = \lim_{n\to \infty} \left(1 + x/n \right) ^n\) Now,with matrix valued argument,the above equation can be written as \(\displaystyle\lim_{k\to\infty}\ [\ I_{\ n\times\ n}\ + \frac{\ i \vec{\phi}\ . \mathbb{D} }{k}] ^k\) \( = \ exp\ [ \ {\ i \ {\vec{\phi}\ . \mathbb{D} ] \) My question is will the relation still hold if \(\mathbb{D}=\mathbb{D}_{1}\ +\mathbb{D}_{2}\) and the two operators D1 and D2 do not commute...? I think it will(as for angular momentum matrices,which are generators of rotation),but how to see it?
What prevents you from just putting D=D1+D1 in the formula? Or are you talking about one of the following? http://mathworld.wolfram.com/TrotterProductFormula.html http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
Matrix addition commutes, always. It is matrix multiplication that in general does not commute. However, given any real square matrix A and a non-negative integer n, A[sup]n[/sup]A=AA[sup]n[/sup]. In other words, A[sup]n[/sup] is unambiguous.
My question was if the result will hold good for the case where D1 and D2 do not commute (in the sense of matrix multiplication).Say,[D1,D2]=cD3 where c is in general a complex number. In the proof of the Gaussformula, we binomially expnad the RHS and taking limit,show that the formula conforms to exp x. But when I am using matrix argument,I am not sure how to write the higher powers of i*phi*D...
Of course the formula holds if you just replace \(D_{1}\) with \(D_{1}+D_{2}\). It's like asking if a formula is still value if you replace x by y+z. Substituting \(D_{1}+D_{2}\) in the place of \(D_{1}\) is just a relabelling, you don't need to know anything about \(D_{1}\)'s relationship with \(D_{2}\) to do that. It's only when you then start manipulating that expression, such as wondering if \(e^{D_{1}+D_{2}} = e^{D_{1}}e^{D_{2}}\), which is only true if \([D_{1},D_{2}]=0\).