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Hi guys I have a problem in finding my error in a calculation, I will be glad if you help me to find the error that I am doing

ok the problem is basically about matrix exponentials, here we go:

A, B, U, P are matrices

n is a natural number

t and T are rational numbers and T=n*t

now in general ## e^{t(A+B)}≠e^{tA}*e^{tB} ##

but can be represented by using The Zassenhaus formula

## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##

one can find the details of the formula from http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula

now I begin by writing the formula

## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##

and then I take the 'n'th power of both sides

## \left(e^{t(A+B)}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

if I define U=A+B I will get

## \left(e^{t U}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

by using the equality of (e^A)^n=e^[aN] i will obtain

## e^{t*n*U}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

by using the definitions T=nt and U=A+B i will get

## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

for the right side I define the P matrix as P=AB-BA

## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*P}*\ldots\right)^n ##

and by using the equality ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ## recursively, I obtain

## e^{T*(A+B)}=e^{t*n*A}*e^{t*n*B}*e^{(t^2/2)*n*P}*\ldots ##

Again by using the definitions of T=tn and P=AB-BA I obtain

## e^{T*(A+B)}=e^{T*A}*e^{T*B}*e^{T*(t/2)*(AB-BA)}*\ldots ##

but, if expand the left side of the equation by using Zassenhaus formula, I end up with

## e^{TA}*e^{TB}*e^{(T^2/2)(AB-BA)}*...≠e^{TA}*e^{TB}*e^{(T*t/2)(AB-BA)}*\ldots ##

which is not clearly equal to the right side; so where is my mistake.

*****************************

As a side note; If you also show the correct version of the calculation I will be glad;

For example; I expect my error is asuuming the equality of ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ##

if this is the mistake, could you show the correct relation?

Thanks for the help

ok the problem is basically about matrix exponentials, here we go:

A, B, U, P are matrices

n is a natural number

t and T are rational numbers and T=n*t

now in general ## e^{t(A+B)}≠e^{tA}*e^{tB} ##

but can be represented by using The Zassenhaus formula

## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##

one can find the details of the formula from http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula

now I begin by writing the formula

## e^{t(A+B)}=e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots ##

and then I take the 'n'th power of both sides

## \left(e^{t(A+B)}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

if I define U=A+B I will get

## \left(e^{t U}\right)^n=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

by using the equality of (e^A)^n=e^[aN] i will obtain

## e^{t*n*U}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

by using the definitions T=nt and U=A+B i will get

## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*(AB-BA)}*\ldots\right)^n ##

for the right side I define the P matrix as P=AB-BA

## e^{T*(A+B)}=\left(e^{tA}*e^{tB}*e^{(t^2/2)*P}*\ldots\right)^n ##

and by using the equality ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ## recursively, I obtain

## e^{T*(A+B)}=e^{t*n*A}*e^{t*n*B}*e^{(t^2/2)*n*P}*\ldots ##

Again by using the definitions of T=tn and P=AB-BA I obtain

## e^{T*(A+B)}=e^{T*A}*e^{T*B}*e^{T*(t/2)*(AB-BA)}*\ldots ##

but, if expand the left side of the equation by using Zassenhaus formula, I end up with

## e^{TA}*e^{TB}*e^{(T^2/2)(AB-BA)}*...≠e^{TA}*e^{TB}*e^{(T*t/2)(AB-BA)}*\ldots ##

which is not clearly equal to the right side; so where is my mistake.

*****************************

As a side note; If you also show the correct version of the calculation I will be glad;

For example; I expect my error is asuuming the equality of ## \left(e^A*e^B\right)^n=e^{(nA)}*e^{(nB)} ##

if this is the mistake, could you show the correct relation?

Thanks for the help

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