It has nothing to do with WPF.
I know this problem. You need to sandwich your string in quotation marks:
string application = @"""E:\Visual Studio 2010\Projects\chess\chess\bin\Debug\chess.exe""";
string app2 = "\"E:\\Visual Studio 2010\\Projects\\chess\\chess\\bin\\Debug\\chess.exe\"";
ProcessStartInfo startInfo = new ProcessStartInfo(application);
Do I have to explain why? A hint: there is a blank space character in your path. If you still did not get it, please ask a follow-up question.
A big warning for you: you cannot make supportable application if you use immediate constants hard-coded in your code. As to path names, it cannot work in portable way in principle. All path strings should always be calculated based on your main executable module of your entry assembly or put in configuration file. Alternatively, the user can specify it using a file dialog. There is no case when a hard-coded value could work, ever.
—SA