nonstandard solutions of the Schwinger-Dyson equations [Archive]

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very_cryptic@hotmail.com

Dec24-04, 06:50 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>While it is well known that Feynman path integrals form a class of\nsolutions to the Schwinger-Dyson equations, they most definitely aren\'t\nthe most general solution.\n\nLet me give an example:\n\nIn the phi to the 4th theory, the SD equation for the generating\nfunctional is\n\n[-i lambda/3! \\partial^3/\\partial J(x)^3+i(box+m^2) \\partial/\\partial\nJ(x)+J(x)]Z[J]=0\n\nLet phi_i(\\vec{x}) be an arbitrary function at t=t_i. Then, for t>t_i,\n\nZ[J]=\\int \\mathcal{D}phi_{phi(t_i)=phi_i} exp{i\\int d^4x 1/2 (\\partial\nphi)^2-m^2/2 phi^2 - lambda/4! phi^4 + J phi}\n\nforms the class of solutions of interest.\n\nHowever, there are infinitely many other classes of solutions. Take for\nexample,\n\nZ[J]=\\int \\mathcal{D}phi_{phi(t_i)=phi_i} exp{i\\int d^4x -1/2 (\\partial\nphi)^2 + m^2/2 phi^2 - lambda/4! phi^4 - i J phi}\n\nThis is also a class of solutions, except that it gives negative\nprobabilities, which isn\'t good.\n\nThere are also classes of solutions which are odd in J, etc.\n\nBut isn\'t it possible that a linear combination of these two solutions\nwill yield positive probabilities? I have to admit I wasn\'t able to\ncheck the nonnegativity condition for all possible expectation values\nof A*A where A is an operator, but among those combinations I have\nchecked, it appears possible there might be linear combinations of\nstandard and nonstandard solutions which MIGHT possibly always satisfy\nthe nonnegativity conditions.\n\nAt any rate, what is the physical meaning of these nonstandard\nsolutions? After all, there are some physicists who work purely with\nSchwinger-Dyson equations instead of with path integrals.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>While it is well known that Feynman path integrals form a class of
solutions to the Schwinger-Dyson equations, they most definitely aren't
the most general solution.

Let me give an example:

In the \phi to the 4th theory, the SD equation for the generating
functional is

[-i \lambda/3! \partial^3/\partial J(x)^3+i(box+m^2) \partial/\partialJ(x)+J(x)]Z[J]=0

Let \phi_i(\vec{x}) be an arbitrary function at t=t_i. Then, for t>t_i,Z[J]=\int \mathcal{D}\phi_{\phi(t_i)=\phi_i} \exp{i\int d^{4x} 1/2 (\partial\phi)^2-m^2/2 \phi^2 - \lambda/4! \phi^4 + J \phi}

forms the class of solutions of interest.

However, there are infinitely many other classes of solutions. Take for
example,

Z[J]=\int \mathcal{D}\phi_{\phi(t_i)=\phi_i} \exp{i\int d^{4x} -1/2 (\partial\phi)^2 + m^2/2 \phi^2 - \lambda/4! \phi^4 - i J \phi}

This is also a class of solutions, except that it gives negative
probabilities, which isn't good.

There are also classes of solutions which are odd in J, etc.

But isn't it possible that a linear combination of these two solutions
will yield positive probabilities? I have to admit I wasn't able to
check the nonnegativity condition for all possible expectation values
of A*A where A is an operator, but among those combinations I have
checked, it appears possible there might be linear combinations of
standard and nonstandard solutions which MIGHT possibly always satisfy
the nonnegativity conditions.

At any rate, what is the physical meaning of these nonstandard
solutions? After all, there are some physicists who work purely with
Schwinger-Dyson equations instead of with path integrals.

very_cryptic@hotmail.com

Dec25-04, 01:56 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>breasts flat (parboil first if using umbilical\ncords so they won?t be tough).\nPlace a slice of ham and cheese on each, along with some minced onion\nthen fold in half, trimming neatly.\nDredge in flour, eggwash, then seasoned breadcrumbs;\nallow to sit for a few minutes.\nSauté in butter and olive oil until golden brown,\nabout 6 minutes on each side.\n\n\n\nShish Kababes\n\nAs old as the hills, this technique has employed seafood, beef, pork, lamb,\npoultry, and vegetables; just about anything can be grilled, and young humans\nare no exception!\n\nHigh quality marinade (Teriyaki and garlic perhaps)\n1 inch cubes of tender meat, preferably from the nursery\nOnions\nbell peppers\nWooden or metal skewers\n\nMarinate the meat overnight.\nGet the grill good and hot while placing meat, vegetables, and\nfruit such as pineapples or cherries on the skewers.\nDon?t be afraid to use a variety of meats.\nGrill to medium rare,\nserve with garlic cous-cous and sautéed asparagus.\nCoffee and sherbet for desert then walnuts, cheese, and port.\nCigars for the gentlemen (and ladies if they so desire)!\n\n\n\nCrock-Pot Crack Baby\n\nWhen the quivering, hopelessly addicted crack baby succumbs to death,\nget him imme\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>breasts flat (parboil first if using umbilical
cords so they won?t be tough).
Place a slice of ham and cheese on each, along with some minced onion
then fold in half, trimming neatly.
Dredge in flour, eggwash, then seasoned breadcrumbs;
allow to sit for a few minutes.
Sauté in butter and olive oil until golden brown,
about 6 minutes on each side.

Shish Kababes

As old as the hills, this technique has employed seafood, beef, pork, lamb,
poultry, and vegetables; just about anything can be grilled, and young humans
are no exception!

High quality marinade (Teriyaki and garlic perhaps)
1 inch cubes of tender meat, preferably from the nursery
Onions
bell peppers
Wooden or metal skewers

Marinate the meat overnight.
Get the grill good and hot while placing meat, vegetables, and
fruit such as pineapples or cherries on the skewers.
Don?t be afraid to use a variety of meats.
Grill to medium rare,
serve with garlic cous-cous and sautéed asparagus.
Coffee and sherbet for desert then walnuts, cheese, and port.
Cigars for the gentlemen (and ladies if they so desire)!

Crock-Pot Crack Baby

When the quivering, hopelessly addicted crack baby succumbs to death,
get him imme

very_cryptic@hotmail.com

Dec25-04, 02:23 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>You are just about there, Pour the sauce over the cutlets,\ntop with parsley, lemon slices and cracked pepper.\nServe with spinach salad, macaroni and cheese (homemade) and iced tea...\n\n\n\nSpaghetti with Real Italian Meatballs\n\nIf you don?t have an expendable bambino on hand,\nyou can use a pound of ground pork instead.\nThe secret to great meatballs, is to use very lean meat.\n\n1 lb. ground flesh; human or pork\n3 lb. ground beef\n1 cup finely chopped onions\n7 - 12 cloves garlic\n1 cup seasoned bread crumbs\n½ cup milk, 2 eggs\nOregano\nbasil\nsalt\npepper\nItalian seasoning, etc.\nTomato gravy (see index)\nFresh or at least freshly cooked spaghetti or other pasta\n\nMix the ground meats together in a large bowl,\nthen mix each of the other ingredients.\nMake balls about the size of a baby?s fist\n(there should be one lying around for reference).\nBake at 400°for about 25 minutes -\nor you could fry them in olive oil.\nPlace the meatballs in the tomato gravy, and simmer for several hours.\nServe on spaghetti.\nAccompany with green salad, garlic bread and red wine.\n\n\n\nNewborn Parmesan\n\nThis classic Sicilian cuisine can easily be turned into Eggplant Parmesan\nIf you are planning a vegetarian meal. Or you could just as well use veal -\nafter all, you have to be careful - Sicilians are touchy about their young\nfamily members...\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>You are just about there, Pour the sauce over the cutlets,
top with parsley, lemon slices and cracked pepper.
Serve with spinach salad, macaroni and cheese (homemade) and iced tea...

Spaghetti with Real Italian Meatballs

If you don?t have an expendable bambino on hand,
you can use a pound of ground pork instead.
The secret to great meatballs, is to use very lean meat.

1 lb. ground flesh; human or pork
3 lb. ground beef
1 cup finely chopped onions
7 - 12 cloves garlic
1 cup seasoned bread crumbs
½ cup milk, 2 eggs
Oregano
basil
salt
pepper
Italian seasoning, etc.
Tomato gravy (see index)
Fresh or at least freshly cooked spaghetti or other pasta

Mix the ground meats together in a large bowl,
then mix each of the other ingredients.
Make balls about the size of a baby?s fist
(there should be one lying around for reference).
Bake at 400°for about 25 minutes -
or you could fry them in olive oil.
Place the meatballs in the tomato gravy, and simmer for several hours.
Serve on spaghetti.
Accompany with green salad, garlic bread and red wine.

Newborn Parmesan

This classic Sicilian cuisine can easily be turned into Eggplant Parmesan
If you are planning a vegetarian meal. Or you could just as well use veal -
after all, you have to be careful - Sicilians are touchy about their young
family members...

very_cryptic@hotmail.com

Dec25-04, 03:26 PM

very_cryptic@hotmail.com

Dec25-04, 03:33 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>heads stuffed with wild rice. Or keep it simple with a\nhearty main course such as stew, lasagna, or meat loaf.\n\nSome suggestions\n\nPre-mie pot pies, beef stew, leg of lamb, stuffed chicken, roast pork spiral ham,\nCranberry pineapple salad, sweet potatoes in butter, vegetable platter, tossed salad with tomato and avocado, parsley new potatoes, spinich cucumber salad, fruit salad\nBran muffins, dinner rolls, soft breadsticks, rice pilaf, croissants\nApple cake with rum sauce, frosted banana nut bread sherbet, home made brownies\nIced tea, water, beer, bloody marys, lemonade, coffee\n\nThe guests select food, beverages, silverware... everything from the buffet table.\nThey move to wherever they are comfortable, and sit with whoever they choose.\nProvide trays so your guests will not spill everything all over your house from\ncarrying too much, nor will they have to make 10 trips back and fourth from the\nservice stations.\n\n\n\nRoast Leg of Amputee\n\nBy all means, substitute lamb or a good beef roast if the haunch\nit is in any way diseased. But sometimes surgeons make mistakes,\nand if a healthy young limb is at hand, then don?t hesitate to cook\nit to perfection!\n\n1 high quality limb, rack, or roast\nPotatoes, carrot\nOil\ncelery\nonions\ngreen onions\nparsley\ngarlic\nsalt, pepper, etc\n2 cups beef stock\n\nMarinate meat (optional, not necessary with better cuts).\nSeason liberally and lace with garlic cloves by making incisions,\nand placing whole cloves deep into the meat.\nGrease a baking pan, and fill with a thick bed of onions,\ncelery, green onions, and parsley.\nPlace roast on top with fat side up.\nPlace uncovered in 500° oven for 20 minutes, reduce oven to 325°.\nBake till medium rare (150°) and let roast rest.\nPour stock o\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>heads stuffed with wild rice. Or keep it simple with a
hearty main course such as stew, lasagna, or meat loaf.

Some suggestions

Pre-mie pot pies, beef stew, leg of lamb, stuffed chicken, roast pork spiral ham,
Cranberry pineapple salad, sweet potatoes in butter, vegetable platter, tossed salad with tomato and avocado, parsley new potatoes, spinich cucumber salad, fruit salad
Bran muffins, dinner rolls, soft breadsticks, rice pilaf, croissants
Apple cake with rum sauce, frosted banana nut bread sherbet, home made brownies
Iced tea, water, beer, bloody marys, lemonade, coffee

The guests select food, beverages, silverware... everything from the buffet table.
They move to wherever they are comfortable, and sit with whoever they choose.
Provide trays so your guests will not spill everything all over your house from
carrying too much, nor will they have to make 10 trips back and fourth from the
service stations.

Roast Leg of Amputee

By all means, substitute lamb or a good beef roast if the haunch
it is in any way diseased. But sometimes surgeons make mistakes,
and if a healthy young limb is at hand, then don?t hesitate to cook
it to perfection!

1 high quality limb, rack, or roast
Potatoes, carrot
Oil
celery
onions
green onions
parsley
garlic
salt, pepper, etc
2 cups beef stock

Marinate meat (optional, not necessary with better cuts).
Season liberally and lace with garlic cloves by making incisions,
and placing whole cloves deep into the meat.
Grease a baking pan, and fill with a thick bed of onions,
celery, green onions, and parsley.
Place roast on top with fat side up.
Place uncovered in 500° oven for 20 minutes, reduce oven to 325°.
Bake till medium rare (150°) and let roast rest.
Pour stock o

very_cryptic@hotmail.com

Dec25-04, 03:35 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>chickens\nfor frying before you go for the real thing.\nWhole chicken is much more efficient and inexpensive than buying pieces.\n\n1 tiny human, cut into pieces\n2 cups flour\nOnion, garlic\nSalt\npepper\ngarlic powder\ncayenne pepper\nhot sauce, etc.\nOil for frying\n\nMix milk, eggs, hot sauce in a bowl, add chopped onion and garlic.\nSeason the meat liberally, and marinate for several hours.\nPlace seasoned flour in a paper or plastic shopping bag,\ndrop pieces in a few a time, shake to coat thoroughly,\nthen deep fry in hot oil (350°) for about 15 minutes.\nDrain and place on paper towels.\n\n\n\nMiscarriage with Mustard Greens\n\nWhy waste it? Otherwise, and in general, use ham or salt pork to season greens.\nThe technique of smothering greens can be used with many vegetables;\ngreen beans work especially well. Meat is not necessary every day, don?t\nbe afraid to alter any dish to vegetarian tastes.\n\n1 premature baby, born dead\nLarge bunch of mustard greens\n2 white onions, 1 cup chopped celery\nVegetable oil (or hog fat)\nSalt, pepper, garlic, etc.\n\nLightly brown onions, celery, garlic and meat in large heavy pot.\nAdd a little water and the greens (which should be thoroughly cleaned and washed).\nSmother slowly for at least 2 hours, adding small amounts of water\nwhen it starts to stick.\nStir frequently.\nWhen ready - serve with rice, grilled smoke\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>chickens
for frying before you go for the real thing.
Whole chicken is much more efficient and inexpensive than buying pieces.

1 tiny human, cut into pieces
2 cups flour
Onion, garlic
Salt
pepper
garlic powder
cayenne pepper
hot sauce, etc.
Oil for frying

Mix milk, eggs, hot sauce in a bowl, add chopped onion and garlic.
Season the meat liberally, and marinate for several hours.
Place seasoned flour in a paper or plastic shopping bag,
drop pieces in a few a time, shake to coat thoroughly,
then deep fry in hot oil (350°) for about 15 minutes.
Drain and place on paper towels.

Miscarriage with Mustard Greens

Why waste it? Otherwise, and in general, use ham or salt pork to season greens.
The technique of smothering greens can be used with many vegetables;
green beans work especially well. Meat is not necessary every day, don?t
be afraid to alter any dish to vegetarian tastes.

1 premature baby, born dead
Large bunch of mustard greens
2 white onions, 1 cup chopped celery
Vegetable oil (or hog fat)
Salt, pepper, garlic, etc.

Lightly brown onions, celery, garlic and meat in large heavy pot.
Add a little water and the greens (which should be thoroughly cleaned and washed).
Smother slowly for at least 2 hours, adding small amounts of water
when it starts to stick.
Stir frequently.
When ready - serve with rice, grilled smoke