Changing variables into light-cone coordinates
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I feel that I am overlooking something, but for the past hour I've been stuck on what should be rather trivial. I have an integral that I need to evaluate and I would like to change the variables I'm integrating over
beginequation
int...dk_1dk_2 rightarrow int...textJacobiancdot dk^+dk^-
endequation
It doesn't really matter what's inside, as I have denoted with "...". The $k^pm$ are given by
beginequation
k^pm=k_1pm k_2.
endequation
Now I would assume, what I have to do is simply write $dk_1$ as
beginequation
dk_1=fracpartial k_1partial k^+dk^++fracpartial k_1partial k^-dk^-,
endequation
and hence similarly for $dk_2$. Expressing $k^1,2$ in terms of $k^pm$ I get
beginequation
k^1,2=frac12(k^+pm k^-)
endequation
Which means that the way I defined $dk_1,2$ gives me
beginalign
dk_1&=frac12(dk^++dk^-)\
dk_2&=frac12(dk^+-dk^-),
endalign
and finally
beginequation
dk_1dk_2=frac14left[(dk^+)^2-(dk^-)^2right],
endequation
which is not at all what I expected to get. Where do I go wrong?
mathematical-physics jacobian change-of-variable
asked Aug 9 at 16:25
Henrikas
697
add a comment |Â
up vote
1
down vote
favorite
I feel that I am overlooking something, but for the past hour I've been stuck on what should be rather trivial. I have an integral that I need to evaluate and I would like to change the variables I'm integrating over
beginequation
int...dk_1dk_2 rightarrow int...textJacobiancdot dk^+dk^-
endequation
It doesn't really matter what's inside, as I have denoted with "...". The $k^pm$ are given by
beginequation
k^pm=k_1pm k_2.
endequation
Now I would assume, what I have to do is simply write $dk_1$ as
beginequation
dk_1=fracpartial k_1partial k^+dk^++fracpartial k_1partial k^-dk^-,
endequation
and hence similarly for $dk_2$. Expressing $k^1,2$ in terms of $k^pm$ I get
beginequation
k^1,2=frac12(k^+pm k^-)
endequation
Which means that the way I defined $dk_1,2$ gives me
beginalign
dk_1&=frac12(dk^++dk^-)\
dk_2&=frac12(dk^+-dk^-),
endalign
and finally
beginequation
dk_1dk_2=frac14left[(dk^+)^2-(dk^-)^2right],
endequation
which is not at all what I expected to get. Where do I go wrong?
mathematical-physics jacobian change-of-variable
asked Aug 9 at 16:25
Henrikas
697
I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I feel that I am overlooking something, but for the past hour I've been stuck on what should be rather trivial. I have an integral that I need to evaluate and I would like to change the variables I'm integrating over
beginequation
int...dk_1dk_2 rightarrow int...textJacobiancdot dk^+dk^-
endequation
It doesn't really matter what's inside, as I have denoted with "...". The $k^pm$ are given by
beginequation
k^pm=k_1pm k_2.
endequation
Now I would assume, what I have to do is simply write $dk_1$ as
beginequation
dk_1=fracpartial k_1partial k^+dk^++fracpartial k_1partial k^-dk^-,
endequation
and hence similarly for $dk_2$. Expressing $k^1,2$ in terms of $k^pm$ I get
beginequation
k^1,2=frac12(k^+pm k^-)
endequation
Which means that the way I defined $dk_1,2$ gives me
beginalign
dk_1&=frac12(dk^++dk^-)\
dk_2&=frac12(dk^+-dk^-),
endalign
and finally
beginequation
dk_1dk_2=frac14left[(dk^+)^2-(dk^-)^2right],
endequation
which is not at all what I expected to get. Where do I go wrong?
mathematical-physics jacobian change-of-variable
asked Aug 9 at 16:25
Henrikas
697
I feel that I am overlooking something, but for the past hour I've been stuck on what should be rather trivial. I have an integral that I need to evaluate and I would like to change the variables I'm integrating over
beginequation
int...dk_1dk_2 rightarrow int...textJacobiancdot dk^+dk^-
endequation
It doesn't really matter what's inside, as I have denoted with "...". The $k^pm$ are given by
beginequation
k^pm=k_1pm k_2.
endequation
Now I would assume, what I have to do is simply write $dk_1$ as
beginequation
dk_1=fracpartial k_1partial k^+dk^++fracpartial k_1partial k^-dk^-,
endequation
and hence similarly for $dk_2$. Expressing $k^1,2$ in terms of $k^pm$ I get
beginequation
k^1,2=frac12(k^+pm k^-)
endequation
Which means that the way I defined $dk_1,2$ gives me
beginalign
dk_1&=frac12(dk^++dk^-)\
dk_2&=frac12(dk^+-dk^-),
endalign
and finally
beginequation
dk_1dk_2=frac14left[(dk^+)^2-(dk^-)^2right],
endequation
which is not at all what I expected to get. Where do I go wrong?
mathematical-physics jacobian change-of-variable
asked Aug 9 at 16:25
Henrikas
697
asked Aug 9 at 16:25
Henrikas
697
asked Aug 9 at 16:25
Henrikas
697
asked Aug 9 at 16:25
Henrikas
697
697
I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00
add a comment |Â
I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00
I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Hmm. Well, your transformation matrix is $mathbff(k^+, k^-)=left[beginmatrixk_1+k_2\ k_1-k_2endmatrixright],$ making your Jacobian equal to $J=left[beginmatrix1 &1\ 1 &-1endmatrixright],$ with determinant $-2$. Hence, $dk^+,dk^-=-2,dk_1,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1,dk_2=-dfrac12,dk^+,dk^-,$ that $-dfrac12,dk^+,dk^-=dfrac14,[(dk^+)^2-(dk^-)^2].$
answered Aug 9 at 17:13
Adrian Keister
3,64821533
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hmm. Well, your transformation matrix is $mathbff(k^+, k^-)=left[beginmatrixk_1+k_2\ k_1-k_2endmatrixright],$ making your Jacobian equal to $J=left[beginmatrix1 &1\ 1 &-1endmatrixright],$ with determinant $-2$. Hence, $dk^+,dk^-=-2,dk_1,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1,dk_2=-dfrac12,dk^+,dk^-,$ that $-dfrac12,dk^+,dk^-=dfrac14,[(dk^+)^2-(dk^-)^2].$
answered Aug 9 at 17:13
Adrian Keister
3,64821533
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
add a comment |Â
up vote
1
down vote
accepted
Hmm. Well, your transformation matrix is $mathbff(k^+, k^-)=left[beginmatrixk_1+k_2\ k_1-k_2endmatrixright],$ making your Jacobian equal to $J=left[beginmatrix1 &1\ 1 &-1endmatrixright],$ with determinant $-2$. Hence, $dk^+,dk^-=-2,dk_1,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1,dk_2=-dfrac12,dk^+,dk^-,$ that $-dfrac12,dk^+,dk^-=dfrac14,[(dk^+)^2-(dk^-)^2].$
answered Aug 9 at 17:13
Adrian Keister
3,64821533
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hmm. Well, your transformation matrix is $mathbff(k^+, k^-)=left[beginmatrixk_1+k_2\ k_1-k_2endmatrixright],$ making your Jacobian equal to $J=left[beginmatrix1 &1\ 1 &-1endmatrixright],$ with determinant $-2$. Hence, $dk^+,dk^-=-2,dk_1,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1,dk_2=-dfrac12,dk^+,dk^-,$ that $-dfrac12,dk^+,dk^-=dfrac14,[(dk^+)^2-(dk^-)^2].$
answered Aug 9 at 17:13
Adrian Keister
3,64821533
Hmm. Well, your transformation matrix is $mathbff(k^+, k^-)=left[beginmatrixk_1+k_2\ k_1-k_2endmatrixright],$ making your Jacobian equal to $J=left[beginmatrix1 &1\ 1 &-1endmatrixright],$ with determinant $-2$. Hence, $dk^+,dk^-=-2,dk_1,dk_2.$ Apparently, for your problem, it happens to be the case that, since $dk_1,dk_2=-dfrac12,dk^+,dk^-,$ that $-dfrac12,dk^+,dk^-=dfrac14,[(dk^+)^2-(dk^-)^2].$
answered Aug 9 at 17:13
Adrian Keister
3,64821533
answered Aug 9 at 17:13
Adrian Keister
3,64821533
answered Aug 9 at 17:13
Adrian Keister
3,64821533
answered Aug 9 at 17:13
Adrian Keister
3,64821533
3,64821533
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
add a comment |Â
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
I guess, in the end, it really was as trivial as I thought. Thanks.
– Henrikas
Aug 9 at 17:46
add a comment |Â
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I don't see anything wrong with your working. What did you expect to get?
– Adrian Keister
Aug 9 at 16:58
As my initial formula suggests I expected to arrive at $dk^+dk^-$
– Henrikas
Aug 9 at 17:00